Is the following statement True or False: " f'(x) exists for all real numbers x and f'(x) is never 0, then f(1) can not be equal to f(5). "?

Question

Charles Arocha · Accepted Answer

It is true.

I'll try to show a line of thought that can lead us to a way to find the answer to the question. The question asks something about #f(1)# and #f(5)#. We know that #f'(x)# exists for all real #x#. (#f# is differentiable at every real #x#.) We also know that differentiability implies continuity, so we can conclude that #f# is continuous at every real number. We know some things about continuous functions and some things about differentiable functions. Again, we are asked something about #f(1)# and #f(5)#, so we might now think about the closed interval #[1,5]#. Since #f# is continuous and differentiable on #[1,5]# we have some theorems that apply to #f#. Two of those theorems involve #f'(x)#, namely Rolle's Theorem and the Mean Value Theorem. Using either theorem, we can conclude that if #f(1)# were equal to #f(5)#, then there would have to be a #c# in #(1,5)# with #f'(c) = 0#. But there is no such #c#, so #f(1)# cannot be equal to #f(5)#.

Charles Anctil · Answer

undefined True.