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 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)

We are tasked with finding points \((a, b)\) where the direction of maximum increase of the function \( f(x, y) = xy^3 \) is aligned with the vector \(\langle 1, 1 \rangle\), and the slope in that direction is \(\sqrt{18}\).

### Step 1: Gradient of \( f(x, y) \)
The direction of maximum increase is given by the gradient of the function \( f(x, y) \).

The partial derivatives of \( f(x, y) = xy^3 \) are:
\[
\frac{\partial f}{\partial x} = y^3
\]
\[
\frac{\partial f}{\partial y} = 3xy^2
\]

So, the gradient of \( f(x, y) \) is:
\[
\nabla f(x, y) = \left( y^3, 3xy^2 \right)
\]

### Step 2: Gradient alignment with \(\langle 1, 1 \rangle\)
The direction of maximum increase is aligned with the vector \(\langle 1, 1 \rangle\), meaning that the gradient \(\nabla f(x, y)\) must be proportional to \(\langle 1, 1 \rangle\).

Thus, we must have:
\[
y^3 = 3xy^2
\]

Simplifying this equation:
\[
y^2(y - 3x) = 0
\]

This gives two possibilities:
1. \( y = 0 \)
2. \( y = 3x \)

### Step 3: Slope of the maximum increase
The slope of the maximum increase is the magnitude of the gradient in the direction of \(\langle 1, 1 \rangle\), which must equal \(\sqrt{18}\).

The directional derivative in the direction of \(\langle 1, 1 \rangle\) is:
\[
D_{\langle 1, 1 \rangle} f(x, y) = \frac{\nabla f(x, y) \cdot \langle 1, 1 \rangle}{\|\langle 1, 1 \rangle\|}
\]

Since \(\|\langle 1, 1 \rangle\| = \sqrt{2}\), we have:
\[
D_{\langle 1, 1 \rangle} f(x, y) = \frac{y^3 + 3xy^2}{\sqrt{2}}
\]

We are given that the slope in this direction is \(\sqrt{18}\), so:
\[
\frac{y^3 + 3xy^2}{\sqrt{2}} = \sqrt{18}
\]

Multiplying both sides by \(\sqrt{2}\):
\[
y^3 + 3xy^2 = 6\sqrt{2}
\]

### Step 4: Analyze the cases
#### Case 1: \( y = 0 \)
If \( y = 0 \), then \( f(x, y) = 0 \), and both partial derivatives are zero:
\[
\nabla f(0, 0) = \langle 0, 0 \rangle
\]
Thus, there is no direction of increase, and this case does not provide any valid points.

#### Case 2: \( y = 3x \)
Substituting \( y = 3x \) into the equation \( y^3 + 3xy^2 = 6\sqrt{2} \):
\[
(3x)^3 + 3x(3x)^2 = 6\sqrt{2}
\]
\[
27x^3 + 27x^3 = 6\sqrt{2}
\]
\[
54x^3 = 6\sqrt{2}
\]
\[
x^3 = \frac{\sqrt{2}}{9}
\]
\[
x = \left( \frac{\sqrt{2}}{9} \right)^{1/3}
\]

Thus, \( y = 3x \), so:
\[
y = 3\left( \frac{\sqrt{2}}{9} \right)^{1/3}
\]

### Conclusion
There are two possible values for \( x \), corresponding to the positive and negative cube roots of \(\frac{\sqrt{2}}{9}\), so there are two points \((a, b)\).

Thus, the correct answer is:
\[
\boxed{d. \text{There exist two points } (a, b).}
\]

Would you like more details or have any questions?

Here are 5 follow-up questions:
1. How is the directional derivative related to the gradient of a function?
2. What is the significance of the vector \(\langle 1, 1 \rangle\) in this problem?
3. Can you explain why we discard the solution \(y = 0\)?
4. How do we interpret the magnitude of the gradient in relation to the slope?
5. Can the approach used here apply to other functions involving products of variables?

**Tip:** When the gradient is zero, it means there is no direction of maximum increase from that point—you're at a critical point!

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. 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).

Explore how to determine points where the direction of maximum increase in the function f(x, y) = xy^3 aligns with vector ⟨1, 1⟩, and the slope equals √18. This problem includes gradient calculations, directional derivatives, and solving multivariable equations, ideal for students in undergraduate calculus.

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