For f(x, y) = x^2 - 12xy + y^2 + 1, find

Question

For f(x, y) = x² - 12xy + y² + 1, find

a) The direction of the maximum rate of change at P(1,1)

b) The direction of the minimum rate of change at P(1, 1)

Answer 1

The maximum rate of change is in the direction of the gradient of the function which is a vector made up of the partial derivative of f with respect to x and the partial derivative of f with respect to y.

∇f= <2x-12y, -12x+2y>

∇f(1,1) =<-10, -10>

The direction of the minimum rate of change is in the opposite direction of the gradient.

<10,10>