Question

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). "?

Answer 1

It is true.

Explanation:

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)`.