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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?

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By A.M-G.M inequality, we have $$e^{-x_1}+e^{-2x_2}=\dfrac{e^{-x_1}}{2}+\dfrac{e^{-x_1}}{2}+e^{-2x_2} \geq 3 \left(\dfrac{e^{-2x_1-2x_2}}{4}\right)^{\frac{1}{3}} \geq 3\left(\dfrac{e^{-2}}{4}\right)^{\frac{1}{3}} \;.$$

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    $\begingroup$ To make the answer complete, worth adding that this is achieved for $x_2 = \frac{1+\log 2}{3}$ and $x_1=1-x_2$. $\endgroup$
    – Clement C.
    Feb 27 '21 at 4:27

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