I am having trouble proving the following inequality for all $x,y>0$ from "The Mathematics of Nonlinear Programming" by Pressini, Uhl. The book states that it follows from the convexity of an appropriate function, but I am hopelessly lost to see the light.
$$\frac{x}{4}+\frac{3y}{4}\leq\sqrt{\frac{e^{x^2}}{4}+\frac{3e^{y^2}}{4}}$$
It's a multiple-step proof. Use convexity of the function $f(x)=-\sqrt x$ to establish $$ \sqrt{\frac14 e^{x^2} +\frac34 e^{y^2}}\ge \frac14 \sqrt{e^{x^2}}+\frac34\sqrt{e^{y^2}}=\frac14e^{x^2/2} + \frac34e^{y^2/2}\;. $$ Next, you have to show that $$e^{x^2/2}\ge x $$ for all $x$, which follows from the inequalities $$ e^t\ge 1+t\ \mbox{and}\ 1+t^2/2\ge t$$ which hold for all $t$.