[Math] find the maximum and minimum value of the function $x^3+y^3-3x-12y+10$

Question

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

Answer 1

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 $$\min_{x>0,y>0}(x^3+y^3-3x-12y+10)=-8$$