Chapter 12.6, Problem 55E

Gradients in three dimensions Consider the following functions f, points P, and unit vectors u.

1. a. Compute the gradient of f and evaluate it at P
2. b. Find the unit vector in the direction of maximum increase of f at P.
3. c. Find the rate of change of the function in the direction of maximum increase at P.
4. d. Find the directional derivative at P in the direction of the given vector.

55.     $f\left(x,y,z\right)=x^2+2y^2+4z^2+10;P\left(1,0,4\right);\langle \frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}\rangle$