Directional derivatives

Question

find the directions in which the functions increase and decrease most rapidly at Po. Then find the derivatives of the functions in these directions.

19. f(x, y) = x² + xy + y²,

Po(-1, 1)

Answer 1

The function f with f(x, y) = x² + x y + y²,decreases most rapidly in the direction opposite of the gradient vector

at the point P₀(-1,1).

∇f = (2x + y) i + (x + 2y ) j∇

∇f (-1,1) = - i + j

Hence the function decreases most rapidly in the direction of the vector v = <1, -1 > at the point P₀(-1,1).

The unit vector in the direction of v is u = v / ||v|| = < 1/√2 , -1 /√2 >

Then the directional derivative of the function f in the direction of the vector u is

D_uf ( -1, 1) = ∇f (-1,1) • u = < -1, 1> • < 1/√2 , -1 /√2 > = -√2