Help would be appreciated.
Consider $x \in (0,1)$ and $f(x)=x^2$ which is convex we want to show that $\mathbb{E}\Big[f(X)\Big] \geq f\Big[\mathbb{E}(X)\Big]$. Therefore, $\mathbb{E}\Big[X^{2}\Big] \geq \Big[\mathbb{E}(X)\Big]^{2}$ i.e. $$\int_{0}^{1} x^{2}\cdot x^{2} \mathrm{d}x = \tfrac{1}{5} \geq \Big[\int_{0}^{1} x \cdot x^{2} \mathrm{d}x \Big]^{2} = \tfrac{1}{4}^{2} = \tfrac{1}{16}$$.
Is my logic correct or have I missed something out by the theorem itself?