Maximize the function ​f(x,y,z) = x^2 + 2y - z^2 subject to the constraints 4x - y = 0 and y + z = 0.

Question

Maximize the function ​f(x,y,z) = x² + 2y - z² subject to the constraints 4x - y = 0 and y + z = 0.

Answer 1

You can use Lagrange multipliers. Alternatively, you can make this a function of one variable. I shall show you this solution.

4x - y = 0, so y = 4x

y+z = 0, so z = -y = -4x

maximize x^2 + 2(4x) - (-4x)^2 =

x^2 + 8x - 16x^2 = 8x - 15x^2

Taking the derivative, 8 - 30x = 0

x = 4/15

y = 4x = 16/15

z = -y = -16/15

x^2 + 2y - z^2 = (4/15)^2 + 2(16/15) - (16/15)^2 =

16/225 + 32/15 - 256/225 =

-240/225 + 32/15 =

-16/15 + 32/15 =

16/15

Incidentally, the second derivative is -30, so this is a maximum.