Problem 39

Question

Find values of $x$ and $y$ such that $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ simultaneously. $$ f(x, y)=x^{2}+4 x y+y^{2}-4 x+16 y+3 $$

Step-by-Step Solution

Verified

Answer

The values of $x$ and $y$ that satisfy $f_{x}(x, y)=0$ and $f_{y}(x, y)=0$ simultaneously are $x = -2$ and $y = -3$.

1Step 1: Differentiate $f(x, y)$ with respect to $x$

The partial derivative of $f$ with respect to $x$ is: $f_{x}(x, y) = 2x + 4y - 4$. Set this equal to 0 to get the equation $2x + 4y - 4 = 0$.

2Step 2: Differentiate $f(x, y)$ with respect to $y$

The partial derivative of $f$ with respect to $y$ is: $f_{y}(x, y) = 4x + 2y + 16$. Set this equal to 0 to get the equation $4x + 2y + 16 = 0$.

3Step 3: Solve the system of equations

The last step is to solve the system of equations $2x + 4y - 4 = 0$ and $4x + 2y + 16 = 0$ simultaneously. This yields solutions for $x$ and $y$, which are $x = -2$ and $y = -3$.

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