# Minimize $z=e^{-x_{1}}+e^{-2 x_{2}}$ Subject to $\quad x_{1}+x_{2} \leq 1 \ , x_{1}, x_{2} \geq 0$ without KKT

I want to minimize $$z=e^{-x_{1}}+e^{-2 x_{2}}$$ subject to $$\quad x_{1}+x_{2} \leq 1 \ , x_{1}, x_{2} \geq 0$$ without using the KKT conditions. The restrain is on regular inequalities like Cauchy or Jensen ( some of the common known and used) I don't know how to proceed. I tried to lower bound and get that bound to equality. Any suggestions?

By A.M-G.M inequality, we have $$e^{-x_1}+e^{-2x_2}=\dfrac{e^{-x_1}}{2}+\dfrac{e^{-x_1}}{2}+e^{-2x_2} \geq 3 \left(\dfrac{e^{-2x_1-2x_2}}{4}\right)^{\frac{1}{3}} \geq 3\left(\dfrac{e^{-2}}{4}\right)^{\frac{1}{3}} \;.$$
• To make the answer complete, worth adding that this is achieved for $x_2 = \frac{1+\log 2}{3}$ and $x_1=1-x_2$. Feb 27 '21 at 4:27