Shape of a Graph            Practice Curve Sketching

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 $f(x)=x^2e^x$. solution




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




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? 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.




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.

 

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