## Optimisation problem using Khun Tucker karush condition and Lagrange multipliers

The function $f(x,y,z)=c−x^2 −y^2 −z^2$ is to be maximized over $\Bbb R^3$
> subject to the constraints $x+y+z≥p$ and $x≤q$ where $c$, $p$ and $q$ are
> positive real constants.
>
> (i) Write down the Lagrangian function for this nonlinear programming
> problem and state the Kuhn-Tucker-Karush (K-T-K) necessary conditions.
>
> (ii) Determine the gradient vector and the Hessian matrix of the
> Lagrangian function of Part (a)(i). Does the K-T-K Sufficiency Theorem
> hold for any optimal point? Support your answer with sound reasoning.
>
> (iii) Find the values of x, y and z at the optimal point,
> distinguishing between the different cases that can arise depending on
> the values of p and q.

i) the ktk conditions are

$\Delta_{x}L = 0$

$\lambda( x+y+z - p) = 0$

$\mu(q-x) = 0$

$\lambda \geq 0$

$\mu \geq 0$

(i) $L (x,y,x, \lambda , \mu) = c−x^2 −y^2 −z^2 + \lambda(x+y+z-p) + \mu(q-x)$

and the constraints must hold at the optimum,

(ii) $\nabla_{x}L =$

$-2x +\lambda -\mu = 0$

$-2y + \lambda = 0$

$-2z + \lambda = 0$

$H = \begin{pmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \end{pmatrix}$

Can someone please help me with part 2 and 3? How do I show this holds for an optimal point and also how would I work out the values of x , y , and z, I can gather that x = y =z