A function $$ f $$ and a point $$ P $$ are given. Let $$ heta $$ correspond to the direction of the directional derivative. Complete parts (a) through (e). $$ f(x, y)=16-3 x^{2}-2 y^{2}, P(2,3) $$ b. Find the angles $$ heta $$ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angle(s) is/are associated with the direction of maximum increase? $$ \frac{5 \pi}{4} $$ (Type any angles in radians between 0 and $$ 2 \pi $$. Type an exact answer, using $$ \pi $$ as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of maximum decrease? $$ \frac{\pi}{4} $$ (Type any angles in radians between 0 and $$ 2 \pi $$. Type an exact answer, using $$ \pi $$ as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of zero change? The angle(s) associated with the directions of zero change is/are (Type any angles in radians between 0 and $$ 2 \pi $$. Type an exact answer, using $$ \pi $$ as needed. Use a comma to separate answers as needed.)

Question

A function $$ f $$ and a point $$ P $$ are given. Let $$ 	heta $$ correspond to the direction of the directional derivative. Complete parts (a) through (e). $$ f(x, y)=16-3 x^{2}-2 y^{2}, P(2,3) $$ b. Find the angles $$ 	heta $$ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change. What angle(s) is/are associated with the direction of maximum increase? $$ \frac{5 \pi}{4} $$ (Type any angles in radians between 0 and $$ 2 \pi $$. Type an exact answer, using $$ \pi $$ as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of maximum decrease? $$ \frac{\pi}{4} $$ (Type any angles in radians between 0 and $$ 2 \pi $$. Type an exact answer, using $$ \pi $$ as needed. Use a comma to separate answers as needed.) What angle(s) is/are associated with the direction of zero change? The angle(s) associated with the directions of zero change is/are (Type any angles in radians between 0 and $$ 2 \pi $$. Type an exact answer, using $$ \pi $$ as needed. Use a comma to separate answers as needed.)

UpStudy AI · Accepted Answer

Let 
$$
\nabla f(x,y) = \left(-6x,\; -4y\right).
$$
At $$ P(2,3) $$, we have
$$
\nabla f(2,3) = \left(-12,\; -12\right).
$$

**Step 1. Maximum Increase Direction**

The maximum increase occurs in the direction of the gradient. The angle $$ \theta $$ of the vector $$\langle -12, -12 \rangle$$ (with respect to the positive $$ x $$-axis) is given by
$$
\theta = \arctan\left(\frac{-12}{-12}\right) = \arctan(1).
$$
Since the vector $$\langle -12,-12 \rangle$$ lies in the third quadrant, we add $$ \pi $$:
$$
\theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}.
$$
Thus, the angle associated with the direction of maximum increase is
$$
\frac{5\pi}{4}.
$$

**Step 2. Maximum Decrease Direction**

The maximum decrease direction is opposite to the gradient. Therefore, add $$ \pi $$ radians to the gradient’s angle:
$$
\theta = \frac{5\pi}{4} + \pi = \frac{9\pi}{4}.
$$
Since angles are taken modulo $$ 2\pi $$, subtract $$ 2\pi $$ to get a value between $$ 0 $$ and $$ 2\pi $$:
$$
\frac{9\pi}{4} - 2\pi = \frac{\pi}{4}.
$$
Thus, the angle associated with the direction of maximum decrease is
$$
\frac{\pi}{4}.
$$

**Step 3. Zero Change Direction**

Zero directional derivative occurs for any unit vector $$ \mathbf{u} = \langle \cos\theta, \sin\theta \rangle $$ that is orthogonal to the gradient $$ \nabla f(2,3) = \langle -12,-12 \rangle $$. The condition for orthogonality is
$$
(-12)(\cos\theta) + (-12)(\sin\theta) = 0.
$$
Dividing both sides by $$-12$$ (which is nonzero) gives:
$$
\cos\theta + \sin\theta = 0.
$$
Solving for $$\theta$$,
$$
\sin\theta = -\cos\theta \quad \Longrightarrow \quad \tan\theta = -1.
$$
The general solution for $$\tan\theta = -1$$ is:
$$
\theta = -\frac{\pi}{4} + k\pi,\quad k \in \mathbb{Z}.
$$
Choosing values between $$ 0 $$ and $$ 2\pi $$:

For $$ k = 1 $$:
$$
\theta = -\frac{\pi}{4} + \pi = \frac{3\pi}{4}.
$$

For $$ k = 2 $$:
$$
\theta = -\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}.
$$

Thus, the angles associated with the directions of zero change are
$$
\frac{3\pi}{4},\; \frac{7\pi}{4}.
$$
The angles associated with the directions of maximum increase, maximum decrease, and zero change are: - **Maximum Increase:** $$ \frac{5\pi}{4} $$ - **Maximum Decrease:** $$ \frac{\pi}{4} $$ - **Zero Change:** $$ \frac{3\pi}{4}, \frac{7\pi}{4} $$

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