Question

Find the range of the function f defined by f(x,y) = 10-x^2-y^2 for all (x,y) for which x^2+y^2 <=9. Sketch two of its level curves.

Answer 1

Find the critical points $f'_x=-2x=0, f'_y=-2y=0$ hence $f(0,0)=10$ ,$f_{max}=10$

On the border of the area $f(x,y)=10-(x^2+y^2)$ hence $f(x,y)\geq1\, \text{and} \, f(x,y)\le10$

hence $f_{min}=1$