Kuhn-Tucker question with two inequality constraints

Question

I've been asked to solve the following problem.

max $10x-5x^2+2y-y^2+25$

subject to

$1-x-y\ge0$

$1-x^2-y^2\ge0$

Is anyone able to solve this by hand? NB: I've been told that the KT assumptions are not valid for this problem, but I do not understand why

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$L=10x-5x^2+2y-y^2+25 + \lambda_1(1-x-y) + \lambda_2(1-x^2-y^2) $

First condition~:

$\partial L/\partial x =10-x-\lambda_1-2\lambda_2$=0

$\partial L/\partial y =2-2y-\lambda_1-2\lambda_2$=0

Second condition:

$\lambda_1^*\ge0,g_1(x,^*y^*)\ge0,\lambda_2^*\ge0,g_2(x,^*y^*)\ge0$

Third condition:

$\lambda_1^*g_1(x,^*y^*)=0$

and

$\lambda_2^*g_2(x,^*y^*)=0$

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I understand that with two constaints there are four cases:

1. both constraints binding

2. both constraints are not binding

3. constraint one is binding and constraint two is not binding

4. constraint two is binding and constraint one is not binding

but after this point i become lost.

Answer 1

Hint.

Attached a plot showing the three stationary points $(a,b,c)$ and as we can observe the point $c$ obeys the KKT conditions.