Max. value of x^2+ny^2+(n+1)z^2

[kinu]

If \(x,y,z\in\mathbb{R}\) and x^2+y^2+z^2=1

then maximum value of \(x^2+ny^2+(n+1)z^2\)

[João P. Ferreira]

Try using Lagrange multipliers

\(\Lambda (x,y,z,\lambda)=x+ny^2+(n+1)z^2-\lambda(x^2+y^2+z^2-1)\)

We derivate

\(\frac{\partial \Lambda}{\partial x}=1-2\lambda x=0\)

\(\frac{\partial \Lambda}{\partial y}=2ny-2\lambda y=0\)

\(\frac{\partial \Lambda}{\partial z}=2(n+1)z-2\lambda z=0\)

\(\frac{\partial \Lambda}{\partial \lambda}=x^2+y^2+z^2-1=0\)

Now you just have to solve these equations

[kinu]

Thanks admin for giving me a solution

I have solved like this way (actually not mine some one explain me)

Anexos

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[João P. Ferreira]

It might work, but I'm not sure...

I would use Lagrange multiplers

You just have to solve those equations... It's not that hard