Find the derivative of f(x,y) =x^2−xy+y^2at the point (x,y) = (2,1) in the direction that f decreases most rapidly.

Answer 1

In a nutshell

F( x, y) = x² -x y + y²

The gradient vector of F being

∇ F = [ ∂F /∂x ] i + [ ∂F /∂y ] j = [ 2 x - y ] i + [ -x + 2 y ] j THEN

∇ F ( 2 ,1 ) = 3 i

Hence to realize maximum decrease we must walk on the direction opposite

of ∇ F ( 2 ,1 ) .

Therefore F decreases more rapidly in the direction of − 3 i

Answer 2

The conclusion is the direction of the directional derivative

increases maximally is when df/dl=(unit vector direction l)dotproduct(gradient vector f)

which for dotproduct also means df/dl=|unit vector|*gradient vector|*cos(angle between the two vectors)

to end the sentence, is maximal when angle =0 so cos(angle)=1

oppositely, cos is -1 when the unit vector points opposite the gradient vector

gradient at (2,1)= CORRECTION (3,0) leads to correction

direction of maximum decrease is the h=-3i+0j vector

since the direction of maximum increase is 3i+0j