# Mean value theorem and bounds

Back studying some math and I am not sure the argument below is rigorous.

Let $$f(.)$$ be a continuously differentiable function taking values on a compact interval. By the mean value theorem:

$$\exists t, f(a) - f(b) = f'(t)(a-b)$$

By continuity and differentiability of $$f$$, then can I conclude below?

$$f(a) - f(b) \geq f'(t+\varepsilon)(a-b) ~ or ~ f(a) - f(b) \geq f'(t-\varepsilon)(a-b)$$

-Is this legitimate? -Do I need a continuous derivative or only continuity and differentiability suffices?

• If $f'$ is increasing or decreasing. – hamam_Abdallah Oct 6 '18 at 19:56