Maximize/minimize $1/3 x^3 + y$ with constraint $x^2 + y^2 = 1$?

Question

I keep running around in circles when I use the Lagrangian multiplier method getting $x = 1/y$

But then when I substitute $(1/y)^2 + y^2 = 1$

I then get $1/y^2 + y^2 = 1$ and this doesn't give me the proper solution. Thanks

Answer 1

$x^2+y^2 = 1$ so we can say $x = \sin \theta, y = \cos \theta$.

Then we need to maximize and minimize $\frac{1}{3 \sin^3 \theta} + \cos \theta$ which can be easily done with calculus.