I want to figure out which of the below is not necessarily correct. We have
(1) $$ E[|x|] \geq |E[x]| $$
This is always correct due to Jensen's inequality and the absolute value function being a convex function.
(2) $$ E[x^4] \geq E[x]^4 $$
The function $f(x) = x^4$ is convex because $f"(x) = 12x^2 \geq 0$. So this is always correct as well by Jensen's inequality.
(3) $$ E[x^2] \geq E[|x|] $$
I don't know how to formally justify this because if I plot $x^2$ vs $|x|$ in my mind, then it's clear (visually) that $E[x^2] < E[|x|]$ if $x \in [0, 1]$. So this one is not always correct. How can I justify this mathematically or is what I stated sufficient justification?
(4) $$ E[\exp x] \geq 1 + E[x] $$
For this one, I think we just simply do a taylor series expansion of $\exp x$ about $x = 0$ and then apply linearity of expectation. So $$ \exp x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \ldots \\ \implies \exp x \geq 1 + x \\ \implies E[\exp x] \geq 1 + E[x] $$ Is this approach correct?
You're correct, that the third one is the one that's not always true; your reasoning is solid too: To get a counterexample take $X$ to be a random variable such that $P(X=1/2)=1$ for instance.
For the last one, I'd just say an extra word about why this works when $x\leq 0$.