Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Find values of \(x\) and \(y\) such that \(f_{x}(x, y)=0\) and \(f_{y}(x, y)=0\) simultaneously. $$ f(x, y)=3 x^{3}-12 x y+y^{3} $$

Step-by-Step Solution

Answer

The solution is \(x = 0\), \(y = 0\) and \(x = 2\), \(y = 2\). These are the values for which \(f_{x}(x, y)=0\) and \(f_{y}(x, y)=0\) simultaneously.

Step 1: Computation of Partial Derivatives

Calculate the partial derivative of the given function \(f(x, y)\) with respect to \(x\) and \(y\). Use the power rule of differentiation which states that the derivative of \(x^n\) with respect to \(x\) is \(n \cdot x^{n−1}\). For \(f_{x}(x, y)\), treat \(y\) as a constant and differentiate \(f(x, y)\) with respect to \(x\). Similarly, for \(f_{y}(x, y)\), treat \(x\) as a constant and differentiate \(f(x, y)\) with respect to \(y\). The derivative of a constant is zero.

Step 2: Solve Equations

Having computed \(f_{x}(x, y)\) and \(f_{y}(x, y)\), set both of these equal to zero. This will give two equations, solve this system of equations to get values of \(x\) and \(y\). It's important to solve for one variable terms of the other variable in one equation, then substitute it into the other equation.

Step 3: Validate Solution

Substitute the obtained values of \(x\) and \(y\) into both \(f_{x}(x, y) = 0\) and \(f_{y}(x, y) = 0\) to verify if they satisfy both equations simultaneously.