MTH 151
Exam 3
Fall 2013
Show all work in a neat and organized fashion.
Clearly indicate your answers.
100 points possible.
x=
−b±
√
b2−4ac
2a
1. (25 pts.) Given:
f(x) = 2x3+15x2−36x
Use calculus methods, showing
all work, to do the following.
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the x-coordinates of all local maximum and local minimum points of f.
(You don't have to find the y-coordinates.)
(c) Find the intervals of concavity for f.
(d) Find the x-coordinates of all points of inflection of f.
(You don't have to find the y-coordinates.)
2. (25 pts.) Given:
f(x) = unknown continuous function, with domain the set of all real numbers
f′(x) =
4x+5
3x2/3
f"(x) =
4x−10
9x5/3
Use calculus methods, showing
all work, to do the following.
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the x-coordinates of all local maximum and local minimum points of f.
(You don't have to find the y-coordinates.)
(c) Find the intervals of concavity for f.
(d) Find the x-coordinates of all points of inflection of f.
(You don't have to find the y-coordinates.)
3. (20 pts.) The graph of the derivative f′ of a function f is shown.
(a) Find the intervals on which f is increasing or decreasing.
(b) Find the x-coordinates of all local maximum and local minimum points of f.
(You don't have to find the y-coordinates.)
(c) Find the intervals of concavity for f.
(d) Find the x-coordinates of all points of inflection of f.
(You don't have to find the y-coordinates.)
4. (15 pts.) Verify that the function satisfies the hypotheses of the
Mean Value Theorem on the given interval [a,b]. Then find all numbers c in (a,b) such that
f′(c)=
f(b)−f(a)
b−a
.
f(x)=x2/3, [0,1]
5. (15 pts.) Set up the following applied problem on a closed interval and
use the Candidates Test (i.e., Closed Interval Method), showing all work, to justify your solution.
A rectangular box has a square base. The edge of the base must be at least 1 ft.
The box has no top, and the total area of its five sides is 192 ft2.
What is the maximum possible volume of such a box?
Optional Bonus Problem. (5 optional bonus points possible)
Find the limit. Support your answer with symbolic/algebraic work.