# 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 $$x>0$$

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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.