Suppose f:RR is bounded and has a continuous derivative. What is right and what
ID: 1941516 • Letter: S
Question
Suppose f:RR is bounded and has a continuous derivative. What is right and what is wrong in the following string of conclusions?
"We want to prove that the set T of all points at which f assumes its (absolute) maximum is closed. Since f is differentiable, it is continuous. Hence it assumes its maximum; that is, T is not empty. Denote by S the set of points at which f'(x)=0. Then TS. On the other hand, if x S, then f'(x)=0; hence f achieves a maximum or a minimum there. If it achieves a maximum, we must have f(x)0. Hence T = S{x|f(x)0}. Since {x|f(x)0} is closed, as is S, T is closed."
Explanation / Answer
since f(x) is differentiable, it means it is continous, but we cant say neccessarily that f'(x)=0 at some point. for example, f(x) = e^x is also continous & differentiable, but it doesnt mean that it has a point where f'(x)=0