Exam Question from multivariable calculus.

Question

This question from a previous multivariable calculus exam.I don't know how to start with this question:

Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that there is a number $s\in (0,1)$ ,such that

$f(x_0+h)-f(x_0)=f'(x_0+sh).h$

Please help with some hint for how to go for the answer to question...

Answer 1

Hint. Use the Mean Value Theorem to the function $$ g(t)=f(x_0+th), \quad t\in [0,1]$. $$ Then $$ f(x_0+h)-f(x_0)=g(1)-g(0)=g'(s), \quad \text{for some}\,\,\,s\in (0,1). $$