Question

Optimize{z=f(x,y)=6x^2-9x-3xy-7y+5y^2 }
using the step-by-step procedure.

Answer 1

Equation given is

$f(x,y) = 6x^{2} - 9x-3xy-7y+5y^{2}$

taking partial derivative with respect to x and y

$\frac{\partial f}{\partial x} = 12x-9-3y$

$\frac{\partial f}{\partial y} = -3x-7+10y$

solving above equation to find x and y

$12x-9-3y=0$ $\implies y = 4x-3$ . . . . . . . (i)

$-3x-7+10y=0$ . . . . . . . (ii)

putting value of y from (i) to (ii),

$-3x-7+10(4x-3) = 0$

$-3x-7 + 40x -30 = 0 \implies x=1$

then from (i),

$y=4(1) - 3 = 1$

taking higher partial derivatives,

$r =\frac{\partial^{2} f}{\partial x^{2} } = 12$

$t= \frac{\partial^{2} f}{\partial y^{2} } = 10$

$s= \frac{\partial^{2} f}{\partial x \partial y } = -3$

Now

$rt-s^{2} = 12*10 - 9 > 0$

So point (1,1) is point of local minima.

Minimum value of f(x,y)

$f(1,1) = 6-9-3-7+5 = -8$