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 given the function \( f(x, y) = xy^3 \) and the following conditions:

1. The direction of maximum increase is aligned with the vector \( \langle 1, 1 \rangle \).
2. The slope in the direction of maximum increase is \( \sqrt{18} \).

Let's analyze this step-by-step.

### Step 1: Gradient of the function
The direction of the maximum increase of a function is given by the gradient \( \nabla f(x, y) \), which is the vector of partial derivatives with respect to \( x \) and \( y \). For \( f(x, y) = xy^3 \), the gradient is:

\[
\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)
\]

Calculating the partial derivatives:

\[
\frac{\partial f}{\partial x} = y^3
\]
\[
\frac{\partial f}{\partial y} = 3xy^2
\]

Thus, the gradient is:

\[
\nabla f(x, y) = \langle y^3, 3xy^2 \rangle
\]

### Step 2: Condition on the direction of the gradient
We are told that the direction of maximum increase is aligned with the vector \( \langle 1, 1 \rangle \). This implies that the gradient vector \( \nabla f(x, y) \) must be proportional to \( \langle 1, 1 \rangle \). In other words, there exists a scalar \( \lambda \) such that:

\[
\nabla f(x, y) = \lambda \langle 1, 1 \rangle
\]

This gives us the system of equations:

\[
y^3 = \lambda
\]
\[
3xy^2 = \lambda
\]

Equating the two expressions for \( \lambda \):

\[
y^3 = 3xy^2
\]

Factoring out \( y^2 \) (assuming \( y \neq 0 \)):

\[
y^2(y - 3x) = 0
\]

So, \( y = 0 \) or \( y = 3x \).

#### Case 1: \( y = 0 \)
Substituting \( y = 0 \) into the partial derivatives gives \( \nabla f(x, 0) = \langle 0, 0 \rangle \), which is not aligned with \( \langle 1, 1 \rangle \). Hence, \( y = 0 \) does not provide any solutions.

#### Case 2: \( y = 3x \)
Substitute \( y = 3x \) into one of the equations for \( \lambda \):

\[
y^3 = (3x)^3 = 27x^3
\]

Thus, \( \lambda = 27x^3 \).

### Step 3: Condition on the slope
The slope in the direction of maximum increase is given by the magnitude of the gradient. We are told that this slope is \( \sqrt{18} \). The magnitude of the gradient is:

\[
|\nabla f(x, y)| = \sqrt{\left( y^3 \right)^2 + \left( 3xy^2 \right)^2}
\]

Substituting \( y = 3x \) into this:

\[
|\nabla f(x, 3x)| = \sqrt{(27x^3)^2 + (9x^3)^2}
\]
\[
|\nabla f(x, 3x)| = \sqrt{729x^6 + 81x^6} = \sqrt{810x^6} = \sqrt{810} |x^3|
\]
\[
|\nabla f(x, 3x)| = 9\sqrt{10} |x^3|
\]

We are told that this equals \( \sqrt{18} \), so:

\[
9\sqrt{10} |x^3| = \sqrt{18}
\]

Solving for \( x \):

\[
9\sqrt{10} |x^3| = \sqrt{18}
\]
\[
|x^3

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⟩ and 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 the function f(x,y)=xy^3 to determine if points (x, y) exist where the gradient is aligned with ⟨1, 1⟩ and the maximum slope is √18. This solution covers key multivariable calculus concepts, including partial derivatives and gradient vectors. Suitable for advanced high school or college-level students.

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