<s>[INST] find the maximum and minimum value of the function $x^3+y^3-3x-12y+10$ find the maximum and minimum value of the function $x^3+y^3-3x-12y+10$? here the question is of $2$ variables and i am not able to solve that ,i know how to solve maximum and  minimum question of 1 variable,so please help me to solve this
 [/INST] For positive variables by AM-GM we obtain:
$$x^3+y^3-3x-12y+10=x^3+1+1+y^3+8+8-3x-12y-8\geq$$
$$\geq3\sqrt[3]{x^3\cdot1\cdot1}+3\sqrt[3]{y^3\cdot8\cdot8}-3x-12y-6=-8.$$
The equality occurs for $x=1$ and $y=2$, which says that 
$$\min_{x>0,y>0}(x^3+y^3-3x-12y+10)=-8$$
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