Question

Find the maximum rate of change of f(x,y) = x2y +2y

at the point (-1,2) and the direction in which it occurs.

Answer 1

Maximum rate of change at the point (-1,2) = √17

Direction is the direction of the gradient

Explanation:

The gradient of a function (scalar or vectorial ) is a vector in the direction of maximum rate of change then

f( x,y ) = x*2y + 2y

grad = δ/δx i + δ/δy j + δ/δz k

grad f(x,y) = [ δ/δx i , δ/δy] = [ 2y , x+2 ]

at the point ( -1 , 2 )

grad f(x,y) = [4 , 1]

| grad f(x,y) | = √ (4)² + (1)² = √17