July 9, 2013
1. on the domain [0,5]
a. The Mean Value Theorem guarantees that there exists a point c at which h’(c) has a certain
value. What is that value?
b. Find the value of c that satisfies the conclusion of the MVT?
a. Find and
b. Find the critical number(s) of f .
, but these values of x are not in f’s domain, so the only critical number of f is e
c. Use the first derivative test to determine whether the critical number is a local maximum, local minimum or neither.
If x<e, ln(x)<1, so f’(x)>0, and if x<e, ln(x)>1 so f’(x)<0. So by the first derivative test x=2 must be a local maximum
d. Use the second derivative test to confirm your answer from part (c).
e. On what interval(s) is f(x) concave up? Concave down?
f is concave up when f’’(x)>0
which is when , and similarly f is concave down when
a. Find. Express your answer in factored form.
b. What are the critical numbers of f ?
c. Use the first derivative test to determine whether each critical number is a local maximum, local minimum, or neither.
So f has a local maximum at -4, a local minimum at -8/5, and neither at x=2.
a. What are the critical numbers of g ?
so the critical numbers are 0,-1,-2
b. Use the second derivative test to determine whether each critical number is a local max or local min.
So g has local minima at x=-2 and 0, and a local maximum at x=-1