Maximize $4x_1+x_2+3x_3$ given the constraint $x_1+x_2+x_3=x$, I used lagrange multiplier and it gave me 3 different values of $\lambda$, what does this indicate?
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3Do you have other constraints on $x_i$'s (for instance non-negativity)? Else the maximum is $\infty$. – Apr 09 '13 at 19:08
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Yes, $x_1,x_2,x_3 $are non-negative – user63219 Apr 09 '13 at 19:22
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The easiest way I see is as follows. We have $$4x_1 + x_2 + 3x_3 = 4(x_1 + x_2 + x_3) - 3x_2 - x_3 = 4x - 3x_2 - x_3$$ Since $x_2,x_3 \geq 0$, we have $ - 3x_2 - x_3 \leq 0$, which gives us $$4x - 3x_2 - x_3 \leq 4x$$Equality holding when $x_1 = x$ and $x_2 = x_3 = 0$.
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Yes, this makes perfect sense, but can you help me with the lagrange method? – user63219 Apr 09 '13 at 20:14