Question

Find the maximum and minimum values of f(x,y)=xy on the ellipse 6x2+y2=5.

Please help!!

Answer 1

first solve ellipse equation for y
\[y = \sqrt{5-6x^{2}}\]
substitute into f(x,y)
\[f(x) = x \sqrt{5-6x^{2}}\]
set derivative equal to 0 to find min/max
\[f'(x) = \sqrt{5-6x^{2}}-\frac{6x^{2}}{\sqrt{5-6x^{2}}} = \frac{5-12x^{2}}{\sqrt{5-6x^{2}}}\]
f'(x) = 0 when numerator is 0
\[5-12x^{2} = 0\]
\[\ x = \pm \sqrt{\frac{5}{12}} \approx \pm 0.6455\]
\[\max \rightarrow f(.6455) \approx 1.02\]
\[\min \rightarrow f(-.6455) \approx -1.02\]