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.