Maximum and Minimum Values

1. Find all critical points of the function: \( \displaystyle{ f(x,y)=2x^2+y^2+2xy+2x+2y-5}\)   solution

2. Find all critical points of  the function: \( \displaystyle{ f(x,y)=2x^3+xy^2+5x^2+y^2+2}\)    solution

3. Use the second derivative test to identify any critical points and determine whether each critical point is a maximum, minimum, saddle point, or none of these for the function  \(\displaystyle{\quad f(x,y) =3x^2+8y^3+12x-12y^2+10 }\)   solution

4. Find the local maximum and minimum values and saddle point(s) of the function.  \(\quad \displaystyle{ f(x,y)=x^3+y^3-3x^2-3y^2-9x+10}\).    solution

5. Find the absolute extrema of the function \( \displaystyle{f(x,y)=2x^3+y^4} \)  on the set \( \displaystyle{D=\{(x,y) | x^2+y^2 \leq 1 \}}\).   solution

6. Find the absolute extrema of the function \(\: \displaystyle{f(x,y)=xy-x-2y+2} \) on the triangular region \(R\) with vertices \( (0,0), \, (4, 0) \, \) and \((0,4)\).   solution   

7. A cardboard box without a lid must have a volume of \(23,328\, cm^3\) . Find the dimensions that minimize the amount of cardboard used.    solution