Find the Maximum/Minimum Value f(x)=2(x-4)^5

Since ${\displaystyle 2}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 2{(x-4)}^5}$ with respect to ${\displaystyle x}$ is ${\displaystyle 2\frac{d}{dx}\left[{(x-4)}^5\right]}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=x^5}$ and ${\displaystyle g\left(x\right)=x-4}$.

Differentiate.

Since ${\displaystyle 10}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 10{(x-4)}^4}$ with respect to ${\displaystyle x}$ is ${\displaystyle 10\frac{d}{dx}\left[{(x-4)}^4\right]}$.

Differentiate using the chain rule, which states that ${\displaystyle \frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]}$ is ${\displaystyle f\prime \left(g\left(x\right)\right)g\prime \left(x\right)}$ where ${\displaystyle f\left(x\right)=x^4}$ and ${\displaystyle g\left(x\right)=x-4}$.

Differentiate.

Find the first derivative.

The first derivative of ${\displaystyle f\left(x\right)}$ with respect to ${\displaystyle x}$ is ${\displaystyle 10{(x-4)}^4}$.

Set the first derivative equal to ${\displaystyle 0}$.

Divide each term in ${\displaystyle 10{(x-4)}^4=0}$ by ${\displaystyle 10}$ and simplify.

Set the ${\displaystyle x-4}$ equal to ${\displaystyle 0}$.

Add ${\displaystyle 4}$ to both sides of the equation.

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.

Subtract ${\displaystyle 4}$ from ${\displaystyle 4}$.

Raising ${\displaystyle 0}$ to any positive power yields ${\displaystyle 0}$.

Multiply ${\displaystyle 40}$ by ${\displaystyle 0}$.

Split ${\displaystyle (-\infty ,\infty )}$ into separate intervals around the ${\displaystyle x}$ values that make the first derivative ${\displaystyle 0}$ or undefined.

Substitute any number, such as ${\displaystyle 0}$, from the interval ${\displaystyle (-\infty ,4)}$ in the first derivative ${\displaystyle 10{(x-4)}^4}$ to check if the result is negative or positive.

Substitute any number, such as ${\displaystyle 6}$, from the interval ${\displaystyle (4,\infty )}$ in the first derivative ${\displaystyle 10{(x-4)}^4}$ to check if the result is negative or positive.

Since the first derivative did not change signs around ${\displaystyle x=4}$, this is not a local maximum or minimum.

No local maxima or minima found for ${\displaystyle f\left(x\right)=2{(x-4)}^5}$.