I have a problem from Karush-Kuhn-Tucker condition

Question

maximize: $xy-x$ and s.t. $x^2+y^2 \leq 9$ and $x \ge 0$

Then, I tried to find points using lagrange method. I used 4 cases and they are $(1)$ $\lambda_1 =0$ and $\lambda_2 = 0$

forrect?

Answer 1

Maximize $xy - x$, given constraints: $x^2 + y^2 \leq 9, x \ge 0$

Taking $xy - x = \lambda (x^2 + y^2 -9)$

Taking derivative wrt $x$ and $y$, we get -

$y-1 = 2 \lambda x$ ...(i)

$x = 2 \lambda y$ ...(ii)

From (ii), $\lambda = \frac{x}{2y}$ and substituting in (i)

we get $y(y-1) = x^2 \implies y(y-1) + y^2 \leq 9$

That gives, $y^2 - \frac{y}{2} \leq \frac{9}{2} \implies (y - \frac{1}{4})^2 \leq \frac{73}{16}$

So $y \leq \frac{\sqrt 73 + 1}{4} \approx 2.386001\, $ (only taking positive values as $x \ge 0$ and we need $xy$ to be positive for max value)

taking max value of $y$, gives $x \approx 1.818516$

And max value of $xy-x \approx 2.520465$