daniel suriya

2022-07-19

Jeffrey Jordon

Move all the expressions to the left side of the equation.

Subtract ${y}^{3}$ from both sides of the equation.

$f\left(x,y\right)-{y}^{3}=-{x}^{3}-2xy+5$

Add ${x}^{3}$ to both sides of the equation.

$f\left(x,y\right)-{y}^{3}+{x}^{3}=-2xy+5$

Add $2xy$ to both sides of the equation.

$f\left(x,y\right)-{y}^{3}+{x}^{3}+2xy=5$

Subtract $5$ from both sides of the equation.

$f\left(x,y\right)-{y}^{3}+{x}^{3}+2xy-5=0$

Find the first derivative.

By the Sum Rule, the derivative of $f\left(x,y\right)-{y}^{3}+{x}^{3}+2xy-5$ with respect to $f$ is $\frac{d}{df}\left[f\left(x,y\right)\right]+\frac{d}{df}\left[-{y}^{3}\right]+\frac{d}{df}\left[{x}^{3}\right]+\frac{d}{df}\left[2xy\right]+\frac{d}{df}\left[-5\right]$.

$f\prime \left(f\right)=\frac{d}{df}\left(f\left(x,y\right)\right)+\frac{d}{df}\left(-{y}^{3}\right)+\frac{d}{df}\left({x}^{3}\right)+\frac{d}{df}\left(2xy\right)+\frac{d}{df}\left(-5\right)$

Evaluate $\frac{d}{df}\left[f\left(x,y\right)\right]$.

$f\prime \left(f\right)=\left(x,y\right)+\frac{d}{df}\left(-{y}^{3}\right)+\frac{d}{df}\left({x}^{3}\right)+\frac{d}{df}\left(2xy\right)+\frac{d}{df}\left(-5\right)$

Differentiate using the Constant Rule.

$f\prime \left(f\right)=\left(x,y\right)+0+0+0+0$

Combine terms.

$f\prime \left(f\right)=\left(x,y\right)$

The first derivative of $f\left(x\right)$ with respect to $x$ is $\left(x,y\right)$.

$\left(x,y\right)$

Set the first derivative equal to $0$.

$\left(x,y\right)=0$

Find the values where the derivative is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

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