Consider the function f(x,y)=xy^3 . Does there exist a point or multiple points (x,y)=(a,b) where the following statements are both true: The direction of maximum increase is aligned with the vector ⟨1,1⟩ The slope in the direction of maximum increase is √18 
Select one:
a. There exists one point (a,b) . b. There exist four points (a,b) . c. There exist no points (a,b) . d. There exist two points (a,b) . e. There exist infinitely many points (a,b)

To determine if there are points \((a, b)\) where the given conditions hold, we will analyze the function \( f(x, y) = xy^3 \) using the gradient and directional derivative concepts.

### Step 1: Gradient of the Function
The gradient of \( f(x, y) = xy^3 \) is:
\[
\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)
\]
Compute the partial derivatives:

- \(\frac{\partial f}{\partial x} = y^3\)
- \(\frac{\partial f}{\partial y} = 3xy^2\)

Thus, the gradient vector is:
\[
\nabla f(x, y) = \langle y^3, 3xy^2 \rangle
\]

### Step 2: Direction of Maximum Increase
The direction of maximum increase is given by the gradient vector \(\nabla f(x, y)\), and we are told that this direction must be aligned with the vector \( \langle 1, 1 \rangle \). For the gradients to be aligned, the ratio of the components of \(\nabla f(x, y)\) must match the ratio of the components of \( \langle 1, 1 \rangle \). That is, we must have:
\[
\frac{y^3}{3xy^2} = \frac{1}{1} \implies \frac{y}{3x} = 1 \implies y = 3x
\]

### Step 3: Slope in the Direction of Maximum Increase
The slope in the direction of maximum increase is the magnitude of the gradient vector. We are given that this magnitude should be \( \sqrt{18} \). The magnitude of the gradient \(\nabla f(x, y)\) is:
\[
|\nabla f(x, y)| = \sqrt{(y^3)^2 + (3xy^2)^2} = \sqrt{y^6 + 9x^2y^4}
\]
Substitute \( y = 3x \) from the previous step:
\[
|\nabla f(x, y)| = \sqrt{(3x)^6 + 9x^2(3x)^4} = \sqrt{729x^6 + 9x^2 \cdot 81x^4} = \sqrt{729x^6 + 729x^6} = \sqrt{1458x^6}
\]
\[
|\nabla f(x, y)| = \sqrt{1458} |x| = \sqrt{18} \cdot |x|
\]
For this to equal \( \sqrt{18} \), we require:
\[
\sqrt{18} \cdot |x| = \sqrt{18} \implies |x| = 1
\]
Thus, \( x = 1 \) or \( x = -1 \).

### Step 4: Find Corresponding \( y \) Values
Since \( y = 3x \), we now substitute the values of \( x \):

- If \( x = 1 \), then \( y = 3(1) = 3 \).
- If \( x = -1 \), then \( y = 3(-1) = -3 \).

### Conclusion:
There are two points where the conditions are satisfied: \((1, 3)\) and \((-1, -3)\).

Thus, the correct answer is:
**d. There exist two points (a, b).**

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Would you like more details or have any questions?

Here are 5 related questions:
1. How do you calculate the gradient for a given function?
2. What is the significance of the direction of maximum increase in a multivariable function?
3. How can the directional derivative be used to find rates of change in specific directions?
4. How do you solve systems involving gradients and specific directions like \(\langle 1,1 \rangle\)?
5. What is the geometric interpretation of the magnitude of the gradient vector?

**Tip:** When solving problems involving gradients, always compute the direction and magnitude separately for deeper insights into the behavior of the function.

Consider the function f(x,y)=xy^3 . Does there exist a point or multiple points (x,y)=(a,b) where the following statements are both true: The direction of maximum increase is aligned with the vector ⟨1,1⟩. The slope in the direction of maximum increase is √18. Select one: a. There exists one point (a,b). b. There exist four points (a,b). c. There exist no points (a,b). d. There exist two points (a,b). e. There exist infinitely many points (a,b).

This problem explores the gradient and directional derivative of the function f(x, y) = xy^3 to determine points where the direction of maximum increase is aligned with vector ⟨1, 1⟩, and the slope is √18. Step-by-step solution with key calculus concepts and formulas.

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