### 1. 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.

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

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

### 4. 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 \( f \).

### 5. 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 \( f \).

### 6. 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 \(f\).

### 7. Use the graph of \(y=f'(x)\) below to answer the following
questions about the function \(f\).
solution

### (a) On what interval(s) is the graph of \(f\) increasing or
decreasing? Justify your answer.

### (b) Find the \(x\)-value(s) at which f has a local maximum or
minimum, and state which is which.

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

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

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

###

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

### b. On what intervals is the graph of \( f \) increasing or decreasing?

### c. Find the \(x\)-value(s) at which \( f \) has a local maximum or minimum, and identify which is which.

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

### e. State the \(x\)-coordinates(s) of inflection point(s) of \( f \), if any.

### 8. The graph of the second derivative \( \, f''\) of a function \(f\) shown. Find the \(x\)-coordinates of the inflection points of \(f\). solution

#

#

#

#

#

#