Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 10 (c)/Solution 1

Evaluate $f(x,y)=xy$ at the critical point and boundary points.

At the interior critical point: $$f(0,0)=0.$$

At the boundary points: $$f(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})=\frac{1}{2},$$ $$f(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})=\frac{1}{2},$$ $$f(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})=-\frac{1}{2},$$ $$f(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})=-\frac{1}{2}.$$

Thus, the absolute maximum value is $\frac{1}{2}$ at $$(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}),(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),$$ and the absolute minimum value is $-\frac{1}{2}$ at $$(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}).$$