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
