Answer:
To find all values of \( x \) and \( y \) such that \( f_x(x,y) = 0 \) and \( f_y(x,y) = 0 \) simultaneously for \( f(x,y) = 5x - 25y + xy \), where \( f_x \) and \( f_y \) denote the partial derivatives of \( f \) with respect to \( x \) and \( y \) respectively, you need to follow these steps:
1. Compute the partial derivative of \( f \) with respect to \( x \), denoted as \( f_x \).
2. Compute the partial derivative of \( f \) with respect to \( y \), denoted as \( f_y \).
3. Set both partial derivatives equal to zero and solve the resulting system of equations for \( x \) and \( y \).
Let's do this: wow
1. Compute \( f_x \):
\[ f_x = \frac{\partial f}{\partial x} = 5 + y \]
2. Compute \( f_y \):
\[ f_y = \frac{\partial f}{\partial y} = -25 + x \]
3. Set both partial derivatives equal to zero and solve the resulting system of equations:
\[ 5 + y = 0 \]
\[ -25 + x = 0 \]
Solving these equations simultaneously:
From the first equation: \( y = -5 \)
From the second equation: \( x = 25 \)
So, the solution is \( (x,y) = (25, -5) \).
