Let $x$ be a real number satisfying $0\leq{x}\leq{\frac{\pi}{2}}$. Prove that $(\frac{\sin x}{x})^3\geq{\cos x}$.
I have a problem I need help. The idea of the proof of this problem is to use Fermat's theorem which is stated as follow
If $f(x)$ is a continuous function on $R$ whose first derivative satisfies $f'(x)\geq{0}$ with $a\leq{x}\leq{b}$ then $\max_{x\in [a,b]} f (x) =f(a)$ and $\min_{x\in [a,b]}f(x)=f(b)$. In the case of $f'(x)\leq{0}$, the opposite is true. Theorem 2: if $x'$ is an extreme point of $f(x)$ then $f'(x')=0$ specifically if $f(x)$ is a continuous function with second derivative, then if $f''(x')\geq{0}$ then $x'$ is the minimum point and $f''(x')\leq{0}$ then $x'$ is the maximum point.