Math 181

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?

2.

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
3.

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.

And, again,

So f has a local maximum at -4, a local minimum at -8/5, and neither at x=2.

4.

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