Find $f_{x}, f_{y}$, and $f_{\lambda}$. The symbol $\lambda$ is the Greek letter lambda. \[ f(x, y, \lambda)=x^{2}+y^{2}-\lambda(8 x+5 y-20) \]
Solution
Step 1 ：Given the function \(f(x, y, \lambda)=x^{2}+y^{2}-\lambda(8 x+5 y-20)\), we are asked to find the partial derivatives of the function with respect to \(x\), \(y\), and \(\lambda\).
Step 2 ：The partial derivative of a function with respect to a variable is the derivative of the function with respect to that variable, treating all other variables as constants.
Step 3 ：For \(f_{x}\), we differentiate \(f\) with respect to \(x\), treating \(y\) and \(\lambda\) as constants. This gives us \(f_{x} = -8\lambda + 2x\).
Step 4 ：For \(f_{y}\), we differentiate \(f\) with respect to \(y\), treating \(x\) and \(\lambda\) as constants. This gives us \(f_{y} = -5\lambda + 2y\).
Step 5 ：For \(f_{\lambda}\), we differentiate \(f\) with respect to \(\lambda\), treating \(x\) and \(y\) as constants. This gives us \(f_{\lambda} = -8x - 5y + 20\).
Step 6 ：Final Answer: The partial derivatives of the function \(f(x, y, \lambda)\) with respect to \(x\), \(y\), and \(\lambda\) are \(f_{x} = \boxed{-8\lambda + 2x}\), \(f_{y} = \boxed{-5\lambda + 2y}\), and \(f_{\lambda} = \boxed{-8x - 5y + 20}\).
