What will be the minimum value of $2x^2 + 2y^2 + 4z^2 - 2xy - 4yz -4x - 2z + 15$?

Question

What will be the minimum value of the expression $$ 2x^2 + 2y^2 + 4z^2 - 2xy - 4yz -4x - 2z + 15? $$

Answer 1

You can use calculus or stick to simple algebra and form some squares by clever grouping.

Hint

$$\begin{align} & 2x^2 + 2y^2 + 5z^2 - 2xy - 4yz -4x - 2z + 15 \\ &= \color{blue}{x^2+y^2-2xy}+\color{red}{y^2+4z^2-4yz}+\color{green}{x^2-4x+4}+\color{purple}{z^2-2z+1}+10\\ &=\color{blue}{(x-y)^2}+\color{red}{(y-2z)^2}+\color{green}{(x-2)^2}+\color{purple}{(z-1)^2}+10 \ \color{orange}{\ge \ldots}\end{align} $$

Answer 2

Hint:

Let $p=2x^2+2y^2+5z^2-2xy-4yz-4x-2z+15$

$\iff2x^2-2x(y+2)+2y^2+5z^2-4yz-2z+15-p=0$

As $x$ is real, the discriminant must be $\ge0$

$$4(y+2)^2\ge8(2y^2+5z^2-4yz-2z+15-p)$$

$$2p\ge3y^2+10z^2-8yz-4y-4z+26$$

The equality occurs if $$x=\dfrac{2(y+2)}4$$

Now let $q=3y^2+10z^2-8yz-4y-4z+26$

$\iff3y^2-4y(1+2z)+10z^2-4z+26-q=0$

$$16(1+z)^2\ge12(10z^2-4z+26-q)$$

$$\iff12q\ge104z^2-80z+296=104\left(z-\dfrac5{13}\right)^2+296-104\cdot\left(\dfrac5{13}\right)^2$$

The equality occurs if $y=\dfrac{4(1+2z)}6$ and $z=\dfrac5{13}$