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.

I think you mean that $f$ is a convex function.
Id est, we need to prove that:
$$\frac{a^4+b^4}{2}\geq\left(\frac{a+b}{2}\right)^4$$ or
$$8(a^4+b^4)\geq a^4+4a^3b+6a^2b^2+4ab^3+b^4$$ or
$$7a^4-4a^3b-6a^2b^2-4ab^3+7b^4\geq0$$ or
$$7a^4-14a^3b+7a^2b^2+10a^3b-20a^2b^2+10ab^3+7a^2b^2-14ab^3+7b^4\geq0$$ or
$$(a-b)^2(7a^2+10ab+7b^2)\geq0$$ or
$$(a-b)^2(2a^2+2b^2+5(a+b)^2)\geq0,$$ which is obvious.
