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

Concave up: ( − ∞ , 0 ) \left(-\infty,0\right) ( − ∞ , 0 ) , ( 1 , ∞ ) \left(1,\infty\right) ( 1 , ∞ ) ; Concave down: ( 0 , 1 ) \left(0,1\right) ( 0 , 1 ) ; Inflection points: x = 0 x=0 x = 0 , x = 1 x=1 x = 1

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

In this problem, we're shown a graph of f prime of x. That's the derivative of f of x. Now we want to use this graph to determine the intervals for which f of x is concave up or concave down and then the location of any inflection point. This is where our concavity changes. Now because here we're given the graph of the derivative of our function, remember that we're not looking here for whether it's curving down or curving up, looking like a smile, looking like a frown. We are looking for what happens with our derivative here. Now we know that when a function is concave up, that means that our derivative will be increasing. When it's concave down, that means our derivative will be decreasing. So that's what we're looking for on the graph here because this is the graph of f prime of x rather than the graph of f of x. So let's go ahead and take a look at our graph from left to right, identifying where it's increasing and where it's decreasing. That will tell us the concavity of our original function f of x. Now here, starting from negative infinity, I can see that my function is going up. It is increasing until we reach x equals 0. So that tells me that my original function, so f of x, is concave up from negative infinity until we reach 0 here. Then my then my derivative begins decreasing. It's going down here. So it's gonna be concave down from that point x equals 0 until here we reach x equals 1 where it starts to go up again. So it's concave down where it's where our graph is decreasing from 0 to 1. Then it begins increasing again, telling us that our original function is then concave up from 1 on to infinity. Now in identifying our inflection points here, this is where our concavity changes. Now specifically when looking at the graph of a derivative, we're looking for where it transitions from increasing to decreasing or decreasing to increasing. Those are going to be the points between all of our intervals here. So we can see at x equals 0, it transitions from increasing to now decreasing. So at x equals 0, we know that we have an inflection point there. We can't state that our inflection point is specifically located at 2 because this is the graph of our derivative. We don't know what our actual function value is at that point. So our inflection point at x equals 0 there. Then Continuing on at x equals 1, we know we have another inflection point because this graph transitions from decreasing to now increasing. So we have a second inflection point located at x equals 1. Remember to always pay attention to what type of graph you're looking at. It may not be the graph of your function, and it instead may be the graph of your function's derivative like it was here. Let us know if you have any questions, and let's get a bit more practice.

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

Concave up: ( − ∞ , 0 ) \left(-\infty,0\right) ( − ∞ , 0 ) , ( 1 , ∞ ) \left(1,\infty\right) ( 1 , ∞ ) ; Concave down: ( 0 , 1 ) \left(0,1\right) ( 0 , 1 ) ; Inflection points: x = 0 x=0 x = 0 , x = 1 x=1 x = 1

Concave up: ( − ∞ , 0 ) \left(-\infty,0\right) ( − ∞ , 0 ) , ( 1 , ∞ ) \left(1,\infty\right) ( 1 , ∞ ) ; Concave down: ( 0 , 1 ) \left(0,1\right) ( 0 , 1 ) ; Inflection points: x = − 1 x=-1 x = − 1 , x = 0 x=0 x = 0 , x = 1 x=1 x = 1

Concave down: ( − 1 , 0 ) \left(-1,0\right) ( − 1 , 0 ) , ( 1 , ∞ ) \left(1,\infty\right) ( 1 , ∞ ) ; Concave up: ( 0 , 1 ) \left(0,1\right) ( 0 , 1 ) ; Inflection points: x = 0 x=0 x = 0 , x = 1 x=1 x = 1

Concave down: ( − 1 , 0 ) \left(-1,0\right) ( − 1 , 0 ) , ( 1 , ∞ ) \left(1,\infty\right) ( 1 , ∞ ) ; Concave up: ( 0 , 1 ) \left(0,1\right) ( 0 , 1 ) ; Inflection points: x = − 1 x=-1 x = − 1 , x = 0 x=0 x = 0 , x = 1 x=1 x = 1

6. Graphical Applications of Derivatives

Concavity

Business Calculus

6. Graphical Applications of Derivatives

Concavity

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Multiple Choice

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.

Concave up: $\left(-\infty,0\right)$ , $\left(1,\infty\right)$ ; Concave down: $\left(0,1\right)$ ; Inflection points: $x=0$ , $x=1$

Concave up: $\left(-\infty,0\right)$ , $\left(1,\infty\right)$ ; Concave down: $\left(0,1\right)$ ; Inflection points: $x=-1$ , $x=0$ , $x=1$

Concave down: $\left(-1,0\right)$ , $\left(1,\infty\right)$ ; Concave up: $\left(0,1\right)$ ; Inflection points: $x=0$ , $x=1$

Concave down: $\left(-1,0\right)$ , $\left(1,\infty\right)$ ; Concave up: $\left(0,1\right)$ ; Inflection points: $x=-1$ , $x=0$ , $x=1$

Verified step by step guidance

Step 1: Recall that the concavity of a function f(x) is determined by the sign of its second derivative, f''(x). If f''(x) > 0, the graph of f(x) is concave up, and if f''(x) < 0, the graph of f(x) is concave down.

Step 2: The graph provided is of f'(x), the first derivative of f(x). To determine concavity, observe where f'(x) is increasing or decreasing. If f'(x) is increasing, f''(x) > 0, and f(x) is concave up. If f'(x) is decreasing, f''(x) < 0, and f(x) is concave down.

Step 3: Analyze the graph of f'(x): From x = -∞ to x = 0, f'(x) is increasing, indicating that f(x) is concave up. From x = 0 to x = 1, f'(x) is decreasing, indicating that f(x) is concave down. From x = 1 to x = ∞, f'(x) is increasing again, indicating that f(x) is concave up.

Step 4: Identify inflection points. Inflection points occur where f''(x) changes sign, which corresponds to where f'(x) changes from increasing to decreasing or vice versa. From the graph, these changes occur at x = 0 and x = 1.

Step 5: Summarize the intervals: f(x) is concave up on (-∞, 0) and (1, ∞), concave down on (0, 1), and has inflection points at x = 0 and x = 1.

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