Question

For f(x, y), find all values of x and y such that fₓ(x, y) = 0 and fᵧ(x, y) = 0 simultaneously. f(x, y) = 16/x + 4/y + xy (x, y) = Find fₓ and fᵧ and evaluate each at the given point.

Answer 1

To find the values of x and y such that fₓ(x, y) = 0 and fᵧ(x, y) = 0 simultaneously, we need to find the partial derivatives of f(x, y) with respect to x and y, set them equal to zero, and solve for x and y.

Step-by-step explanation:

To find all values of x and y such that fₓ(x, y) = 0 and fᵧ(x, y) = 0 simultaneously, we need to find the partial derivatives of f(x, y) with respect to x (fₓ) and y (fᵧ), and then set them equal to zero and solve for x and y. Let's start by finding fₓ:

fₓ(x, y) = d/dx (16/x + 4/y + xy)

Using the power rule of differentiation, we can find that fₓ(x, y) = -16/x² + y

Now, let's find fᵧ:

fᵧ(x, y) = d/dy (16/x + 4/y + xy)

Using the power rule of differentiation, we can find that fᵧ(x, y) = -4/y² + x

Setting fₓ and fᵧ equal to zero and solving for x and y, we get:

-16/x² + y = 0

-4/y² + x = 0

Solving these two equations simultaneously will give us the values of x and y that satisfy the given conditions.