Finding minimum and maximum points subject to the constraint $x^2 +y^2 \leq 1$- Is this solvable?

I am trying find the maximum and minimum values for the absolute value or magnitude of the below function subject to the constraint $x^2 +y^2 \leq 1$:$$f(\theta, \phi, x, y) =(\alpha x+\beta y)\left(x^2 +y^2 -2(\alpha x+\beta y)^2\right)$$

Here, $\alpha = \sin \theta \cos \phi$ and $\beta =\sin \theta \sin \phi$.

My intuition:

1. The minimum value is zero and it corresponds to $\alpha =\beta =0$.
2. The maximum value correspond to to $\alpha= 0, \beta =1$ or $\alpha= 1, \beta =0$.

I used numerical simulations to come to this inuitions. Can someone help me formally derive this?