minimize $(1/2)x^TPx+q^tx+r$ subject to $x^tx \le 1$. The usage of the Lagrangian is not allowed.
I view it as a quadratic equation minimisation task. The way I see it I should take the gradient made it equal to zero and find the optimal point $x^{*}=-P^{1/2}q$. Now we want that $|P^{-1/2}q| \le 1$. What to do next?