<s>[INST] How to prove $f(x)=x^4$ is concave up by definition? I know $f(x)=x^4$ is concave up, by calculating its second derivative. However, how to prove that $f(x)=x^4$ is concave up by definition, say $f(\frac{x_1+x_2}{2})<(1/2)f(x_1)+(1/2)f(x_2)$ for all $x_1,~x_2$? I'd tried binomial theorem, but can't get anything.
 [/INST] As an intermediate step, use the inequality
$$ \frac{x^2 + y^2}{2} \ge \left(\frac{x + y}{2}\right)^2 $$
which is relatively easy to prove.
This gives
$$ \frac{x^4 + y^4}{2} = \frac{(x^2)^2 + (y^2)^2}{2} \ge \left(\frac{x^2 + y^2}{2}\right)^2 \ge \left(\left(\frac{x + y}{2}\right)^2\right)^2 = \left(\frac{x + y}{2}\right)^4. $$

Alternatively, using the "usual" definition of convexity, we have
$$ [(1 - \lambda)x^2 + \lambda y^2] - [(1 - \lambda)x + \lambda y]^2 = (1 - \lambda)\lambda (x - y)^2 \ge 0. $$
Being a quadratic form, this is an easy enough sum of squares to compute. One can do it by hand even. Then we do what we did above:
$$ (1 - \lambda)x^4 + \lambda y^4 \ge [(1 - \lambda)x^2 + \lambda y^2]^2 \ge [(1 - \lambda)x + \lambda y]^4 . $$
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