Mathematics 524 Homework #1

1. Page 31 #1

2. Recall Rolle’s theorem: If f is differentiable on the interval [a, b] and f(a) = f(b) = 0 then there is a point ξ between a and b so that f′(ξ) = 0. Construct an argument (complete with pictures) that shows this is true along the following lines:

(a) There is a point ξ between a and b that is a either a maximizer of a minimizer for f on [a, b]. (Please include some pictures that show what is going on.)

(b) At the point ξ we have f′(ξ) = 0 which concludes the argu-

ment.

3. Recall the mean value theorem: If f is differentiable on the interval [a, b] then then for any two distinct points points x_{0}, x_{1} ∈ [x_{0}, x_{1}] there is a point ξ between x_{0} and x_{1} so that

f(x_{1}) − f(x_{0}) = f′(ξ)(x_{1} − x_{0}).

Construct and argument, which will use Rolle’s theorem, for the mean value theorem along the following lines:

(a) Define a new function F by

F (x) = f(x) − f(x_{1})\frac{x − x_{0}}{x_{1} − x_{0}} − f(x_{0})\frac{x_{1} − x}{x_{1} − x_{0}}

then F (x_{0}) = F (x_{1}) = 0.

(b) There is a point ξ between x_{0} and x_{1} so that F′(ξ) = 0. (Hint: Rolle’s theorem.)

(c) Use the definition of F and show that the formula F′(ξ) can be rewritten to give the statement of the mean value theorem.