Question

1. True or False. If f(x) is continuous on a closed interval, then it is enough to look
at the points where f '(x) = 0 in order to find its absolute maxima and minima. Be
prepared to justify your answer.

2.

A racer is running back and forth along a straight path. He finishes the race at
the place where he began.

True or False. There had to be at least one moment, other than the beginning and
the end of the race, when he ”stopped” (i.e., his speed was 0). Be prepared to give a
proof or counterexample.

Answer 1

1. False.

The interval is closed we can't find f'(x) at the endpoints. Hence we need to look at the endpoints as well for absolute maxima and minima.

2 True.

Let f(x) be the displacement. f(x) is a continuous function, hence we can apply the Mean Value Theorem.

The velocity f'(x) is zero at the end of the race, so by the Mean Value Theorem, it must be zero at at least one other point.

f(b) f (a) f' (c) (l

a and b are the starting and ending point of the race, .a = b