Given a point x y and the gradient vector nabla f fx fy of

Given a point (x, y) and the gradient vector nabla f = (f_x, f_y) of a nice function z = f(x, y). At this given point (x, y), find two directions (each specified by a unit vector u) in which the directional derivative (which is a number) is the largest and smallest (not in absolute value). In what direction is the magnitude of the directional derivative zero?

Solution

directional derivative is largest along same direction of grad(f).

directional derivative is smallest along opposite direction of grad(f)

directional derivative is zero along perpendiular direction of grad(f)

since
Duf(x, y) = |f(x, y)| cos
where is an angle between f and u,

If the point (x, y) is fixed, then grad f(x,y) is fixed
cos equals 1 when f(x, y) and u have the same
direction,
equals 1 when f(x, y) and u have opposite directions and is a straight angle,
and is
zero when f(x, y) and u are perpendicular so that is a right angle