Concavity: Videos & Practice Problems

The concavity of a function is determined by the sign of its second derivative. A positive second derivative indicates the function is concave up, resembling a smile, while a negative second derivative indicates concave down, resembling a frown. Inflection points occur where the second derivative equals zero or does not exist, marking a change in concavity. To analyze concavity without a graph, find potential inflection points by setting the second derivative to zero, then test intervals to determine the sign of the second derivative, establishing where the function is concave up or down.

Determining Concavity from the Graph of f

Understanding the concavity of a function is essential in analyzing its behavior. The concavity is determined by the sign of the second derivative of the function. When the second derivative is positive, the function is said to be concave up, resembling a smiley face on a graph. Conversely, when the second derivative is negative, the function is concave down, resembling a frowning face. This visual representation aids in recognizing the concavity of a function.

The first derivative of a function indicates its slope or rate of change. The second derivative, therefore, represents the rate of change of the first derivative, essentially describing how the slope itself is changing. For instance, if the slope is increasing, the second derivative is positive, indicating concavity upwards. If the slope is decreasing, the second derivative is negative, indicating concavity downwards.

When analyzing a graph, tangent lines can be drawn to observe the behavior of the slope. In regions where the function is concave up, the tangent lines lie below the curve, while in concave down regions, the tangent lines lie above the curve. A critical point in this analysis is the inflection point, where the function changes concavity. At an inflection point, the second derivative is either zero or does not exist, similar to critical points for the first derivative.

To identify intervals of concavity and points of inflection, one can examine the graph from left to right. For example, if a function starts off concave down and transitions to concave up, the point where this change occurs is an inflection point. This process can be repeated to find all intervals of concavity and their corresponding inflection points.

In summary, the concavity of a function provides insight into its behavior, and understanding the relationship between the second derivative and concavity is crucial for deeper mathematical analysis.

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

• A. Concave up: $\left(-\infty,0\right)$, $\left(1,\infty\right)$; Concave down: $\left(0,1\right)$; Inflection points: $\left(0,0\right)$, $\left(1,1\right)$
• B. Concave up: $\left(-\infty,0\right)$; Concave down: $\left(0,\infty\right)$; Inflection points: $\left(0,0\right)$
• C. Concave down: $\left(-1,0\right)$, $\left(1,\infty\right)$; Concave up: $\left(0,1\right)$; Inflection points: $\left(0,0\right)$, $\left(1,1\right)$
• D. Concave up: $\left(-1,0\right)$, $\left(1,\infty\right)$; Concave down: $\left(0,1\right)$; Inflection points: $\left(-1,1\right)$, $\left(0,0\right)$, $\left(1,1\right)$

Answer: (A)

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

• A. Concave up: $\left(-2,-1\right)$, $\left(2,4\right)$; Concave down: $\left(-1,1\right)$, $\left(1,2\right)$, $\left(4,5\right)$; Inflection points: $\left(-1,0\right)$, $\left(1,0\right)$, $\left(2,-1\right)$, $\left(4,3\right)$
• B. Concave up: $\left(-2,-1\right)$, $\left(2,4\right)$; Concave down: $\left(-1,2\right)$, $\left(4,5\right)$; Inflection points: $\left(0,1\right)$, $\left(1,0\right)$, $\left(2,-1\right)$, $\left(4,3\right)$
• C. Concave up: $\left(-2,1\right)$, $\left(2,4\right)$; Concave down: $\left(1,2\right)$, $\left(4,5\right)$; Inflection points: $\left(1,0\right)$, $\left(2,-1\right)$, $\left(4,3\right)$
• D. Concave up: $\left(-2,0\right)$, $\left(2,4\right)$; Concave down: $\left(0,2\right)$, $\left(4,5\right)$; Inflection points: $\left(0,1\right)$, $\left(1,0\right)$, $\left(2,-1\right)$, $\left(4,3\right)$

Answer: (C)

For the following graph, find the open intervals for which the function is concave up or concave down. Identify any inflection points.

• A. Concave down: $\left(0,\frac{\pi}{2}\right)$; Concave up: $\left(\frac{\pi}{2},\pi\right)$; Inflection pt: $\left(\frac{\pi}{2},0\right)$
• B. Concave down: $\left(0,\frac{\pi}{2}\right)$; Concave up: $\left(\frac{\pi}{2},\pi\right)$; Inflection pt: $\left(0,\frac{\pi}{2}\right)$
• C. Concave down: $\left(0,\frac{\pi}{2}\right)$; Concave up: $\left(\frac{\pi}{2},\pi\right)$; Inflection pts: $\left(\frac{\pi}{4},5\right)$, $\left(\frac{3\pi}{4},-5\right)$
• D. Concave down: $\left(\frac{\pi}{4},\frac{3\pi}{4}\right)$ ; Concave up: $\left(0,\frac{\pi}{4}\right)$, $\left(\frac{3\pi}{4},\pi\right)$; Inflection pts: $\left(\frac{\pi}{4},5\right)$, $\left(\frac{3\pi}{4},-5\right)$

Answer: (A)

Determining Concavity from the Graph of f' or f''

Understanding the concavity of a function is essential in calculus, as it provides insights into the behavior of the function's graph. Concavity refers to the direction in which a function curves. A function is said to be concave up when it curves upwards, resembling a cup that can hold water, and concave down when it curves downwards, resembling an upside-down cup.

The key to determining concavity lies in the second derivative of the function. If the second derivative, denoted as \( f''(x) \), is positive (\( f''(x) > 0 \)), the function \( f(x) \) is concave up. Conversely, if the second derivative is negative (\( f''(x) < 0 \)), the function is concave down. This relationship allows us to analyze the concavity of a function even when we are given the graph of its second derivative.

When examining the graph of the second derivative, the position relative to the x-axis is crucial. If the graph is above the x-axis, the second derivative is positive, indicating that the original function is concave up. If the graph is below the x-axis, the second derivative is negative, indicating that the original function is concave down. The point where the second derivative changes sign marks an inflection point, where the concavity of the original function changes.

Similarly, the first derivative, denoted as \( f'(x) \), also provides information about concavity. The second derivative represents the rate of change of the first derivative. If the first derivative is increasing, the original function is concave up; if it is decreasing, the original function is concave down. Thus, by analyzing the graph of the first derivative, one can also determine the concavity of the original function.

In summary, to assess the concavity of a function based on its derivatives, remember the following:

• If \( f''(x) > 0 \), the function is concave up.
• If \( f''(x) < 0 \), the function is concave down.
• If \( f'(x) \) is increasing, the function is concave up; if decreasing, the function is concave down.
• Inflection points occur where the second derivative changes sign.

By mastering these concepts, you will be well-equipped to analyze the concavity of functions using their derivatives, enhancing your understanding of their graphical behavior.

The graph of $f^{\prime}\left(x\right)$is shown below. Use the graph to determine the intervals for which $f\left(x\right)$is concave up or concave down and the location of any inflection points.

• A. Concave up: $\left(-\infty,0\right)$, $\left(1,\infty\right)$; Concave down: $\left(0,1\right)$; Inflection points: $x=0$, $x=1$
• B. Concave up: $\left(-\infty,0\right)$, $\left(1,\infty\right)$; Concave down: $\left(0,1\right)$; Inflection points: $x=-1$, $x=0$, $x=1$
• C. Concave down: $\left(-1,0\right)$ , $\left(1,\infty\right)$; Concave up: $\left(0,1\right)$; Inflection points: $x=0$, $x=1$
• D. Concave down: $\left(-1,0\right)$ , $\left(1,\infty\right)$; Concave up: $\left(0,1\right)$; Inflection points: $x=-1$, $x=0$, $x=1$

Answer: (A)

The graph of $f''(x)$ is shown below. Use the graph to determine the intervals for which $f\left(x\right)$is concave up or concave down and the location of any inflection points.

• A. Concave down: $\left(-1,0\right)$, $\left(2,\infty\right)$; Concave up: $\left(0,2\right)$; Inflection points: $x=0$, $x=2$
• B. Concave down: $\left(-1,0\right)$, $\left(2,\infty\right)$; Concave up: $\left(0,2\right)$; Inflection points: $x=-1$, $x=0$, $x=2$
• C. Concave up: $\left(-\infty,0\right)$, $\left(2,\infty\right)$; Concave down: $\left(0,2\right)$; Inflection points: $x=0$, $x=2$
• D. Concave up: $\left(-\infty,-1\right)$, $\left(0,\infty\right)$; Concave down: $\left(-1,0\right)$; Inflection points: $x=-1$, $x=0$

Answer: (C)

Determining Concavity from the Graph of f' or f'' Example 1

Understanding the behavior of a function involves analyzing its increasing and decreasing intervals, concavity, and points of inflection. A function is considered increasing when it moves upward as you move from left to right, while it is decreasing when it moves downward. To identify these intervals, we can examine the graph of the function.

For instance, if we start at an endpoint, say at \( x = -5 \), and observe that the function decreases until \( x = -4 \), we can conclude that the function is decreasing in the interval \((-5, -4)\). Following this, if the function increases until \( x = -2 \), we note the increasing interval as \((-4, -2)\). Continuing this process, we can identify all intervals of increasing and decreasing behavior throughout the function.

Next, we analyze the concavity of the function. A function is concave up if it resembles a smile (the curve opens upwards) and concave down if it resembles a frown (the curve opens downwards). This can also be determined by examining the position of tangent lines relative to the curve. If the tangent line lies below the curve, the function is concave up; if it lies above, the function is concave down. For example, if the function is concave up from \( x = -5 \) to \( x = -3 \) and then concave down from \( x = -3 \) to \( x = 0 \), we can clearly delineate these intervals.

Points of inflection occur where the concavity changes. These points can be identified at the boundaries of the concavity intervals. For instance, if the function transitions from concave down to concave up at \( x = 0 \), this point is noted as an inflection point. The coordinates of these points can be expressed as pairs, such as \((-3, 1)\) for the first inflection point, \((0, -2)\) for the second, and so on.

It is crucial to differentiate between increasing/decreasing behavior and concavity. The first derivative of a function indicates whether it is increasing (positive derivative) or decreasing (negative derivative), while the second derivative indicates concavity. A function can be increasing while being concave down, or decreasing while being concave up, highlighting the importance of understanding these concepts separately.

Determining Concavity Given a Function

Understanding concavity is essential in calculus, as it provides insights into the behavior of functions. The concavity of a function is determined by the sign of its second derivative. If the second derivative is positive, the function is concave up, resembling a "smile," while a negative second derivative indicates the function is concave down, resembling a "frown." An important aspect to note is that concavity can change at points known as inflection points, where the second derivative is either zero or does not exist.

To analyze concavity without a graph, one must follow a systematic approach. First, identify potential inflection points by finding the second derivative of the function and setting it equal to zero. For example, consider the function \( f(x) = 2x^3 + 12x^2 + 9x - 4 \). The first derivative, calculated using the power rule, is \( f'(x) = 6x^2 + 24x \). Differentiating again gives the second derivative:

$$ f''(x) = 12x + 24 $$

This can be factored as:

$$ f''(x) = 12(x + 2) $$

Setting the second derivative equal to zero to find potential inflection points yields:

$$ 12(x + 2) = 0 \implies x = -2 $$

Next, create a sign chart based on the potential inflection point. This divides the number line into intervals: \( (-\infty, -2) \) and \( (-2, \infty) \). To determine the sign of the second derivative in each interval, select test values. For the interval \( (-\infty, -2) \), choose \( x = -3 \):

$$ f''(-3) = 12(-3 + 2) = 12(-1) = -12 $$

Since the second derivative is negative, the function is concave down on \( (-\infty, -2) \). For the interval \( (-2, \infty) \), choose \( x = 0 \):

$$ f''(0) = 12(0 + 2) = 12(2) = 24 $$

Here, the second derivative is positive, indicating the function is concave up on \( (-2, \infty) \). Thus, the function is concave down from \( (-\infty, -2) \) and concave up from \( (-2, \infty) \).

Since the concavity changes at \( x = -2 \), this point is confirmed as an inflection point. Understanding these concepts allows for a deeper analysis of functions and their graphical representations, enhancing problem-solving skills in calculus.

Determining Concavity Given A Function Example 2

To determine the concavity of the function \( f(x) = 9x^{\frac{1}{5}} \), we first need to analyze its second derivative. The concavity of a function is determined by the sign of its second derivative: if the second derivative is positive, the function is concave up; if it is negative, the function is concave down. Inflection points occur where the concavity changes, which is identified by finding where the second derivative is equal to zero or does not exist.

We start by calculating the first derivative using the power rule. The first derivative is given by:

\[f'(x) = \frac{9}{5} x^{-\frac{4}{5}}.\]

Next, we differentiate again to find the second derivative:

\[f''(x) = -\frac{36}{25} x^{-\frac{9}{5}}.\]

To express this in a more usable form, we rewrite it as:

\[f''(x) = -\frac{36}{25} \frac{1}{x^{\frac{9}{5}}}.\]

Now, we identify potential inflection points by setting the second derivative equal to zero or determining where it does not exist. Since the numerator is a constant and does not equal zero, we focus on where the second derivative does not exist, which occurs when the denominator is zero. This happens at:

\[x = 0.\]

Thus, \( x = 0 \) is our only potential inflection point. We can now create a sign chart to analyze the intervals of concavity. The sign chart is divided into two intervals: \( (-\infty, 0) \) and \( (0, \infty) \).

For the interval \( (-\infty, 0) \), we can test a value, such as \( x = -1 \):

\[f''(-1) = -\frac{36}{25} \frac{1}{(-1)^{\frac{9}{5}}} = -\frac{36}{25} \cdot (-1) = \frac{36}{25} > 0.\]

This indicates that the function is concave up on the interval \( (-\infty, 0) \).

For the interval \( (0, \infty) \), we can test a value, such as \( x = 1 \):

\[f''(1) = -\frac{36}{25} \frac{1}{1^{\frac{9}{5}}} = -\frac{36}{25} < 0.\]

This indicates that the function is concave down on the interval \( (0, \infty) \).

From this analysis, we conclude that the function is concave up on \( (-\infty, 0) \) and concave down on \( (0, \infty) \). Since the concavity changes at \( x = 0 \), this point is classified as an inflection point.

To find the coordinates of the inflection point, we evaluate the original function at \( x = 0 \):

\[f(0) = 9 \cdot 0^{\frac{1}{5}} = 0.\]

Thus, the inflection point is located at the ordered pair \( (0, 0) \).

In summary, the function \( f(x) = 9x^{\frac{1}{5}} \) is concave up on the interval \( (-\infty, 0) \), concave down on the interval \( (0, \infty) \), and has an inflection point at \( (0, 0) \).

Determining Concavity Given a Function Example 3

To analyze the concavity of the polynomial function \( f(x) = 5x^2 - 2x + 24 \), we begin by determining its first and second derivatives. The first derivative, calculated using the power rule, is given by:

\( f'(x) = 10x - 2 \)

Next, we find the second derivative by differentiating the first derivative:

\( f''(x) = 10 \)

To identify potential inflection points, we look for values where the second derivative is equal to zero or does not exist. In this case, since \( f''(x) = 10 \) is a constant positive value, it is never equal to zero and exists everywhere. This indicates that there are no points where the concavity changes.

A function is considered concave up when its second derivative is positive. Since \( f''(x) = 10 > 0 \) for all \( x \), we conclude that the function is concave up on the entire interval from negative infinity to positive infinity:

Concave up: \( (-\infty, \infty) \)

As there are no changes in concavity, the function has no inflection points. This behavior aligns with the characteristics of a quadratic function that opens upwards, as indicated by the positive coefficient of the \( x^2 \) term.

Determine the intervals for which the function is concave up or concave down. State the inflection points.
$f(x)=2x(x+9)^2$

• A. Concave down: $\left(-\infty,\infty\right)$ ; No Inflection Points
• B. Concave down: $\left(-\infty,-6\right)$; Concave up: $\left(-6,\infty\right)$; Inflection pt: $\left(-6,-108\right)$
• C. Concave down: $\left(-\infty,6\right)$; Concave up: $\left(6,\infty\right)$; Inflection pt: $\left(6,2700\right)$
• D. Concave down: $\left(-\infty,0\right)$; Concave up: $\left(0,\infty\right)$; Inflection pt: $\left(0,0\right)$

Answer: (B)

Determine the intervals for which the function is concave up or concave down. State the inflection points.
$f(x)=\frac{7}{2x-9}$

• A. Concave down: $\left(-\infty,\infty\right)$; No Inflection Points
• B. Concave up: $\left(-\infty,\infty\right)$; No Inflection Points
• C. Concave down: $\left(-\infty,\frac92\right)$ ; Concave up: $\left(\frac92,\infty\right)$; Inflection point: $\left(\frac92,0\right)$
• D. Concave down: $\left(-\infty,\frac92\right)$ ; Concave up: $\left(\frac92,\infty\right)$; No Inflection Points

Answer: (D)

Determine the intervals for which the function is concave up or concave down. State the inflection points. 
$f(x)=2\sin x+3x$; $0$ < $x$ < $2\pi$

• A. Concave down: $\left(0,\pi\right)$; concave up: $\left(\pi,2\pi\right)$; Inflection pt: $\left(\pi,3\pi\right)$
• B. Concave down: $\left(0,\pi\right)$; concave up: $\left(\pi,2\pi\right)$; Inflection pt: $\left(\pi,2+3\pi\right)$
• C. Concave down: $\left(0,\frac{\pi}{2}\right)$, $\left(\pi,\frac{3\pi}{2}\right)$; concave up: $\left(\frac{\pi}{2},\pi\right)$, $\left(\frac{3\pi}{2},2\pi\right)$; Inflection pts: $\left(\frac{\pi}{2},\frac{4+3\pi}{2}\right)$, $\left(\pi,3\pi\right)$, $\left(\frac{3\pi}{2},\frac{-4+9\pi}{2}\right)$
• D. Concave up: $\left(0,2\pi\right)$; No inflection pt

Answer: (A)

Here’s what students ask on this topic:

What is concavity in calculus?

Concavity in calculus refers to the direction in which a function curves. If a function is concave up, it curves upwards like a smile, and if it is concave down, it curves downwards like a frown. The concavity of a function is determined by the sign of its second derivative. A positive second derivative indicates concave up, while a negative second derivative indicates concave down. Inflection points occur where the second derivative equals zero or does not exist, marking a change in concavity.

How do you determine the concavity of a function?

To determine the concavity of a function, follow these steps: 1) Find the second derivative of the function. 2) Identify potential inflection points by setting the second derivative equal to zero or finding where it does not exist. 3) Create intervals around these points. 4) Choose test values within each interval and plug them into the second derivative. 5) If the second derivative is positive in an interval, the function is concave up there; if negative, the function is concave down.

What is an inflection point?

An inflection point is a point on the graph of a function where the concavity changes. This means the function changes from being concave up to concave down, or vice versa. At an inflection point, the second derivative of the function is either zero or does not exist. Identifying inflection points is crucial for understanding the overall shape and behavior of the function's graph.

How do you find inflection points?

To find inflection points, follow these steps: 1) Compute the second derivative of the function. 2) Set the second derivative equal to zero and solve for the x-values. 3) Determine where the second derivative does not exist. 4) These x-values are potential inflection points. 5) Verify by checking the sign of the second derivative on either side of these points to confirm a change in concavity.

What is the relationship between the first and second derivatives in determining concavity?

The first derivative of a function gives the slope or rate of change of the function. The second derivative provides the rate of change of the first derivative, indicating how the slope is changing. If the second derivative is positive, the slope is increasing, and the function is concave up. If the second derivative is negative, the slope is decreasing, and the function is concave down. Thus, the second derivative directly determines the concavity of the function.