(L15) Concavity, Second Derivative, and Applications in Business Calculus

Concavity and the Second Derivative

Definitions and Notation

The second derivative of a function f is the derivative of f'. It measures the rate of change of the rate of change, and is denoted by several notations:

• $f''$

• $f''(x)$

• $\frac{d^2y}{dx^2}$

• $\frac{d^2}{dx^2}[f(x)]$

• $D^2[f(x)]$

Similarly, the third derivative is the derivative of f'', denoted by f''', f'''(x), etc.

The nth derivative of f is the derivative of the (n – 1)th derivative, denoted by $f^{(n)}(x)$.

Note: Parentheses around the n distinguish $f^{(n)}(x)$ from $[f(x)]^n$.

Derivatives of Polynomials

• The derivative of a polynomial of degree m is a polynomial of degree m – 1.

• The second derivative is a polynomial of degree m – 2.

• All higher derivatives are zero once the degree drops to zero.

For example, for $f(x) = a_m x^m + a_{m-1} x^{m-1} + \cdots + a_1 x + a_0$:

• First derivative: $f'(x)$ is degree $m-1$

• Second derivative: $f''(x)$ is degree $m-2$

• nth derivative: $f^{(n)}(x)$ is degree $m-n$

Examples: Computing Higher Derivatives

• Example 1: $f(x) = e^x + 4x^4 - x^3 + x^2 - x + 1$ $f'(x) = e^x + 16x^3 - 3x^2 + 2x - 1$ $f''(x) = e^x + 48x^2 - 6x + 2$ $f^{(100)}(x) = e^x$ (since all polynomial terms vanish after enough derivatives)

• Example 2: $f(x) = e^{2x}$ $f'(x) = 2e^{2x}$ $f''(x) = 4e^{2x}$ $f^{(n)}(x) = 2^n e^{2x}$ $f^{(100)}(x) = 2^{100} e^{2x}$

• Example 3: $f(x) = x^{-1}$ $f'(x) = -x^{-2}$ $f''(x) = 2x^{-3}$ $f^{(n)}(x) = (-1)^n n! x^{-n-1}$ $f^{(100)}(x) = \frac{100!}{x^{101}}$

Velocity, Speed, and Acceleration

Definitions

• If $y = s(t)$ is the position of an object at time $t$, then the velocity is $v(t) = s'(t)$.

• The speed is $|v(t)|$, the absolute value of velocity.

• The acceleration is $a(t) = v'(t) = s''(t)$.

Interpreting Signs of Velocity and Acceleration

• If $v(t) > 0$ and $a(t) > 0$: Object moves in positive direction and speeds up.

• If $v(t) > 0$ and $a(t) < 0$: Object moves in positive direction and slows down.

• If $v(t) < 0$ and $a(t) > 0$: Object moves in negative direction and slows down.

• If $v(t) < 0$ and $a(t) < 0$: Object moves in negative direction and speeds up.

Explanation: The object is speeding up when the speed, $|v(t)|$, is increasing. It is slowing down when the speed is decreasing.

Graphical Representation

The following graphs illustrate the four situations above, showing $y = s(t)$ versus $t$ for different signs of $v$ and $a$.

• When $v$ and $a$ have the same sign, the object speeds up.

• When $v$ and $a$ have opposite signs, the object slows down.

Concavity and Inflection Points

Concavity

• If the graph of $f$ lies above all its tangent lines on an interval $I$, it is concave upward (C.U.).

• If the graph of $f$ lies below all its tangent lines on $I$, it is concave downward (C.D.).

Inflection Points

• A point $P = (c, f(c))$ is an inflection point if $f$ is continuous at $c$ and the graph changes concavity at $P$ (from C.U. to C.D. or vice versa).

• If the graph has a tangent line at an inflection point, it crosses the tangent line there.

Concavity and the Second Derivative

• If $f''(x) > 0$ for all $x$ in an interval $I$, then $f$ is C.U. on $I$.

• If $f''(x) < 0$ for all $x$ in $I$, then $f$ is C.D. on $I$.

Example: Concavity Test

Find intervals where $f(x) = x^4 - 4x^3$ is C.U. and C.D., and find inflection points.

• $f'(x) = 4x^3 - 12x^2$

• $f''(x) = 12x^2 - 24x = 12x(x-2)$

• Set $f''(x) = 0$: $x = 0$ or $x = 2$ (possible inflection points)

• Test intervals: $(-\infty, 0)$, $(0, 2)$, $(2, \infty)$

• $f''(x) > 0$ on $(-\infty, 0)$ and $(2, \infty)$ (C.U.)

• $f''(x) < 0$ on $(0, 2)$ (C.D.)

• Inflection points: $(0, 0)$ and $(2, -16)$

Second Derivative Test

Purpose

The second derivative test helps determine if a function has a local maximum or minimum at a critical point.

Test Statement

• Suppose $f'(c) = 0$ and $f''$ is continuous near $c$.

• If $f''(c) > 0$, $f$ has a local minimum at $x = c$.

• If $f''(c) < 0$, $f$ has a local maximum at $x = c$.

• If $f''(c) = 0$, the test is inconclusive; use the first derivative test instead.

Example: Second Derivative Test

• For $f(x) = x^4 - 4x^3$, $f'(x) = 4x^3 - 12x^2$, $f''(x) = 12x^2 - 24x$

• At $x = 3$, $f'(3) = 0$, $f''(3) = 12 \, (3)^2 - 24 \, (3) = 108 - 72 = 36$

• Since $f''(3) > 0$, $f$ has a local minimum at $x = 3$.

Application to Economics: Point of Diminishing Returns

Inflection Points in Business Context

If $y = f(x)$ describes the revenue $y$ obtained from some amount $x$ spent in advertisement, investment, or production, then the graph of $f$ usually has an inflection point $(c, f(c))$ where the concavity changes from upward to downward. This point is called the point of diminishing returns.

• At the point of diminishing returns, additional investment yields less additional revenue than before.

• Mathematically, it is where $f''(x)$ changes sign.

*Additional info: The notes include both typed and handwritten explanations, with examples and diagrams illustrating the concepts of concavity, inflection points, and the second derivative test. The economic application is a standard topic in business calculus, connecting calculus concepts to real-world business decisions.*