Find the value of the maximum rate of change of f at the given point.
A) 0
B) 12
D) 13

Question

Find the value of the maximum rate of change of f at the given point. 
A) 0 
B) 12 
D) 13

Quizplus · Accepted Answer

The maximum rate of change of a function at a point is given by the magnitude of its gradient (i.e. the magnitude of the vector formed by its partial derivatives with respect to all its variables). Therefore, to find the maximum rate of change of f at the given point, we need to take the partial derivatives of f with respect to x and y, and evaluate them at the given point.

f(x,y) = 3xy - x^3 - y^3

∂f/∂x = 3y - 3x^2     (partial derivative with respect to x)

∂f/∂y = 3x - 3y^2     (partial derivative with respect to y)

At the given point (1,2), we have

∂f/∂x = 3(2) - 3(1)^2 = 3

∂f/∂y = 3(1) - 3(2)^2 = -9

So the gradient vector is (3,-9), and the magnitude of this vector is

|grad f| = sqrt(3^2 + (-9)^2) = sqrt(90) = 3sqrt(10)

Therefore, the maximum rate of change of f at the given point is 3sqrt(10).

Find the Value of the Maximum Rate of Change of F