# Find all values of x and y such that f_{x}(x, y) = 0 and f_{y}(x, y) = 0 simultaneously.

f(x, y) = 15x^{3} - 3xy + 15y^{3}.

**Solution:**

Given: Function f(x, y) = 15x^{3} - 3xy + 15y^{3}.

f_{x}(x, y) ⇒ This means differentiate the function with respect to x

d/dx (f(x, y)) = 15(3)x^{2} - 3y

= 45x^{2} - 3y

Given that f_{x}(x, y) = 0

⇒ 45x^{2} - 3y = 0

f_{y}(x, y) ⇒ This means differentiate the function with respect to y

= d/dy (fy(x, y)) = 15(3)y^{2} - 3x

= 45y^{2} - 3x

Given that fy(x, y) = 0

⇒ 45y^{2} -3x = 0

As both functions occurr simultaneously, equate with each other

45x^{2} - 3y = 45y^{2} - 3x

45x^{2} - 45y^{2} = 3y - 3x

45(x^{2} - y^{2}) = 3(x - y)

We know that (x^{2} - y^{2}) = (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.

## Find all values of x and y such that f_{x}(x, y) = 0 and f_{y}(x, y) = 0 simultaneously.

f(x, y) = 15x^{3} - 3xy + 15y^{3}.

**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.

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