Find all values of x and y such that f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously.
f(x, y) = 15x³ - 3xy + 15y³.

Solution:

Given: Function f(x, y) = 15x³ - 3xy + 15y³.

f_x(x, y) ⇒ This means differentiate the function with respect to x

d/dx (f(x, y)) = 15(3)x² - 3y

= 45x² - 3y

Given that f_x(x, y) = 0

⇒ 45x² - 3y = 0

f_y(x, y) ⇒ This means differentiate the function with respect to y

= d/dy (fy(x, y)) = 15(3)y² - 3x

= 45y² - 3x

Given that fy(x, y) = 0

⇒ 45y² -3x = 0

As both functions occurr simultaneously, equate with each other

45x² - 3y = 45y² - 3x

45x² - 45y² = 3y - 3x

45(x² - y²) = 3(x - y)

We know that (x² - y²) = (x + y)(x - y)

15(x - y)(x + y) = (x - y)

x + y = 1/15

Thus, the values of x, y can be anything that sum up to 1/15.

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Find all values of x and y such that f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously.
f(x, y) = 15x³ - 3xy + 15y³.

Summary:

All the values of x and y such that f_x(x, y) = 0 and f_y(x, y) = 0 simultaneously can be anything that sum up to 1/15.