# Use mean value theorem to prove the following

Use the mean value theorem to prove that:

$$\cos(x)>1-\frac{x^2}{2}$$ for all $$x>0$$

-

## 1 Answer

I thought there may be better way than my answer... But I'll stick on my answer.

apply mean value theorem on $f(x)= \frac{x^2}{2}+ \cos x$

since $f(0)=1$, we have some $x_0 \in (0,x)$ such that

$\frac{\frac{x^2}{2}+ \cos x -1}{x} = x_0 - \sin x_0 >0$

since $x>0$, we get the results.

-