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. Findand

 

 

 

b. Find the critical number(s) of f .

, but these values of x are not in fs 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