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