Maximum value of function $f(x,y) = xy^2$ with constraint $x^2 + y^2 = 8$

Question

Can you help me to find the maximum value of $f(x,y) = xy^2$ with the constraint that the x and y coordinates must satisfy $g(x,y) = x^2 + y^2 = 8$

How do I start with this?

Answer 1

Lagrange multipliers

$$\nabla f=\lambda\nabla g$$ $$(y^2,2xy)=\lambda(2x,2y)$$

$2xy=\lambda2y$ gives $x=\lambda$ then from $x^2+y^2=8$ together withe $y^2=\lambda2x$ one have $$x^2-8+2\lambda x=x^2-8+2 x^2=3x^2-8=0$$ so $$x=\pm\sqrt{\frac{8}{3}}$$ and $$y^2=8-x^2=8-\frac{8}{3}=\frac{16}{3}$$ so $$y=\pm\frac{4}{\sqrt3}$$ To find the maximum you have to evaluate the function at these 4 values and compare. Immediately you have that $x>0$ and the function reaches maximum at $$(x,y)=\left(\sqrt{\frac{8}3},\pm\frac{4}{\sqrt3}\right)$$

Answer 2

HINT: Use Lagrangian multipliers. $$f(x,y)= \lambda \nabla g(x,y)$$