5. Graphical Applications of Derivatives

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = (x − 1)²(x + 2)

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(-3) exist?

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = 1− 4/x², x ≠ 0

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

g(x) = √(2x − x²)

Use ƒ' and ƒ" to complete parts (a) and (b). 

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x⁹/9 + 3x⁵ - 16x

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of f?

f′(x) = x(x − 1)

Identify the open intervals on which the function is increasing or decreasing.

$f(x)=x^{2/3}(4-x)$

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(3) exist?

86. Use a derivative to show that g(x)=√(x² + ln x) is one-to-one.

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

c. At what points, if any, does f assume local maximum or minimum values?

f′(x) = (x − 1)²(x + 2)

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = 3x + tan x

First Derivative Test

a. Locate the critical points of f.

b. Use the First Derivative Test to locate the local maximum and minimum values.

c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).

f(x) = x²/(x² - 1) on [-4,4]

Use ƒ' and ƒ" to complete parts (a) and (b). 

a. Find the intervals on which f is increasing and the intervals on which it is decreasing.

b. Find the intervals on which f is concave up and the intervals on which it is concave down.

ƒ(x) = x√(x +9)

Analyzing Functions from Derivatives

Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

b. On what open intervals is f increasing or decreasing?

f′(x) = (x − 1)(x + 2)(x − 3)

In Exercises 121–124, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function?

123. y=(4/5)x^5+16x^2-25