Shape of a Graph – Math 160

Consider \( f(x) = x^3 + 3x^2 - 9x + 10 \). Answer the following using calculus. solution

(a) Find the intervals on which \( f \) is increasing or decreasing.

(b) Find the \(x\)-values where \( f \) attains its local maximum and minimum values.
Find the local maximum and local minimum values of \( f(x) = x^2 e^x \). solution
Consider the function \( f(x)=5x^{1/3} - x^{5/3} \). Find the intervals where \( f \) is increasing or decreasing. Also find the local maxima and minima. solution
Let \( f(x) = x^3 + 3x^2 - 24x + 5 \). Answer the following. solution

a. Find all the critical numbers of \( f \).

b. Find the intervals on which \( f \) is increasing or decreasing.

c. Find the local maximum and minimum values of \( f \).

d. Find the intervals on which \( f \) is concave up or down.

e. Find the inflection points of \( f \).
Consider \( f(x)=2x^3 - 3x^2 \). Answer the following using calculus. solution

a. Find the intervals on which \( f \) is concave up or down.

b. Find the inflection points of \( f \).
Consider the function \( f(x)=x + x^2 - x^3 \). Answer the following using calculus. solution

1. Find the intervals on which \( f \) is concave up or concave down.

2. Find the \(x\)-coordinate(s) of inflection point(s) of \( f \).
(Two problems) Use the graph of \( y=f'(x) \) below to answer the following questions about the function \( f \). Solution of graph A Solution of graph B

(A)

(B)

(a) On what interval(s) is the graph of \( f \) increasing or decreasing?

(b) Find the \(x\)-value(s) where \( f \) has a local maximum or minimum.

(c) On which interval(s) is the graph of \( f \) concave up or concave down?

(d) State the \(x\)-coordinate(s) of inflection point(s), if any.
The graph of the derivative function \( y=f'(x) \) is given below. Answer the following questions about \( f(x) \). solution

a. Find all the critical numbers of \( f \).

b. On what intervals is \( f \) increasing or decreasing?

c. Find the \(x\)-value(s) where \( f \) has a local maximum or minimum.

d. On which interval(s) is \( f \) concave up or concave down?

e. State the inflection point(s), if any.
The graph of the second derivative \( f''(x) \) is shown. Find the \(x\)-coordinates of the inflection points of \( f \). solution

Consider \( f(x) = x^3 + 3x^2 - 9x + 10 \). Answer the following using calculus. solution

(a) Find the intervals on which \( f \) is increasing or decreasing.

(b) Find the \(x\)-values where \( f \) attains its local maximum and minimum values.

Find the local maximum and local minimum values of \( f(x) = x^2 e^x \). solution

Consider the function \( f(x)=5x^{1/3} - x^{5/3} \). Find the intervals where \( f \) is increasing or decreasing. Also find the local maxima and minima. solution

Let \( f(x) = x^3 + 3x^2 - 24x + 5 \). Answer the following. solution

a. Find all the critical numbers of \( f \).

b. Find the intervals on which \( f \) is increasing or decreasing.

c. Find the local maximum and minimum values of \( f \).

d. Find the intervals on which \( f \) is concave up or down.

e. Find the inflection points of \( f \).

Consider \( f(x)=2x^3 - 3x^2 \). Answer the following using calculus. solution

a. Find the intervals on which \( f \) is concave up or down.

b. Find the inflection points of \( f \).

Consider the function \( f(x)=x + x^2 - x^3 \). Answer the following using calculus. solution

1. Find the intervals on which \( f \) is concave up or concave down.

2. Find the \(x\)-coordinate(s) of inflection point(s) of \( f \).

(Two problems) Use the graph of \( y=f'(x) \) below to answer the following questions about the function \( f \). Solution of graph A Solution of graph B

(a) On what interval(s) is the graph of \( f \) increasing or decreasing?

(b) Find the \(x\)-value(s) where \( f \) has a local maximum or minimum.

(c) On which interval(s) is the graph of \( f \) concave up or concave down?

(d) State the \(x\)-coordinate(s) of inflection point(s), if any.

The graph of the derivative function \( y=f'(x) \) is given below. Answer the following questions about \( f(x) \). solution

a. Find all the critical numbers of \( f \).

b. On what intervals is \( f \) increasing or decreasing?

c. Find the \(x\)-value(s) where \( f \) has a local maximum or minimum.

d. On which interval(s) is \( f \) concave up or concave down?

e. State the inflection point(s), if any.

The graph of the second derivative \( f''(x) \) is shown. Find the \(x\)-coordinates of the inflection points of \( f \). solution