The relationship between expectation and standard deviation

Question

Let $X$ be a real valued random variable with finite expectation $E(X)$ and standard deviation $D(X)$. Then $(D(X))^2\leq E(X^2)$

I started to think like following: $$D(X) = E(X^2) - (E(X))^2\\(D(X))^2 = (E(X^2))^2 - 2E(X^2)(E(X))^2 + (E(X))^4$$ But how to cintinue I dont know.

Answer 1

You should have $D(X)^2 = E(X^2) - E(X)^2$ which trivially implies $D(X)^2 \leq E(X^2)$.