Millersville University Name Answer Key
Department of Mathematics
MATH 161, Calculus I , Quiz 7
November 3, 2006
Please answer the following questions. Your answers will be evaluated on their correctness,
completeness, and use of mathematical concepts we have covered. Please show all work and
write out your work neatly. Answers without supporting work will receive no credit.
1. For the function f(x) = x5โ5x,
(a) find the intervals on which the function is increasing and decreasing,
For this information we must examine the first derivative.
fโฒ(x) = 5x4โ5
= 5(x4โ1)
= 5(x2โ1)(x2+ 1)
= 5(xโ1)(x+ 1)(x2+ 1)
Thus f(x) is increasing on the interval (โโ,โ1) S(1,โ) and decreasing on the
interval (โ1,1).
(b) find the x-coordinates of the local extrema (if any),
The critical numbers of fare x=ยฑ1. By the first derivative test, fhas a local
maximum at (โ1, f (โ1)) = (โ1,4) and a local minimum at (1, f (1)) = (1,โ4).
(c) find the intervals where the function is concave up and concave down,
For this information we must examine the second derivative, fโฒโฒ(x) = 20x3. The
graph of the function fis concave down on the interval (โโ,0) and concave up
on the interval (0,โ).
(d) find the x-coordinates of the points of inflection (if any),
The concavity of the graph of fchanges at x= 0, thus the only point of inflection
occurs at (0, f (0)) = (0,0).
(e) sketch a plausible graph of f(x) on the axes below.
