# Finding the maximum with multivariable differantiation

$f\left(x_1,x_2\right)=x_1^3+3x_2^3-9x_1x_2$

Find the maximum given that $x_1,x_2≥1$

Just some added info: I know that I have to find the first order and second order partial derivatives; however when I try to find the critical points for $x_1$ or $x_2$ I always get weird numbers. Any help appreciated.

• Yea something like that. But for some reason I plug it into the other partial, and I don't get 0. So either I suck at solving for $x_1,x_2$ or I did the partial wrong. – EconDude Oct 3 '16 at 21:58
• The maximum is clearly at $x_1 \rightarrow \infty$ and $x_2 \rightarrow \infty$. No wonder you're getting "weird" numbers! – David G. Stork Oct 3 '16 at 22:03
• No it should be 0,0 and 3^2/3,3^1/3 – EconDude Oct 3 '16 at 22:05
• But it just seems weird so I was asking haha – EconDude Oct 3 '16 at 22:05
• Yes I get that too, but for some reason when I plugged it back in to check my work it would give me something off. I must have made an addition or subtraction error somewhere idk, maybe my writing is just too sloppy LOL – EconDude Oct 3 '16 at 22:07

## 1 Answer

Hint: $f(t,t) = 4t^3 - 9t^2.$ Does $4t^3 - 9t^2$ have a finite maximum on $[1,\infty)?$