Chapter 12.6, Problem 38E

Interpreting directional derivatives A function f and a point P are given. Let  θ correspond to the direction of the directional derivative.

1. a. Find the gradient and evaluate it at P.
2. b. Find the angles θ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change.
3. c. Write the directional derivative at P as a function of θ; call this function g.
4. d. Find the value of θ that maximizes g(θ) and find the maximum value.
5. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient.

38.     $f\left(x,y\right)=\ln \left(1+2x^2+3y^2\right);P\left(\frac{3}{4}-\sqrt{3}\right)$