$$$$

The following function will be analyzed:

$x^{5} \left ( x + 1 \right ) \left ( x - 1 \right )$

The following are the local minimum and maximum:

$\text{Click "Calculate" to begin.}$

The local minimum and maximum are points on a graph where the function reaches the smallest or largest value in a specific neighborhood. The local minimum is the smallest value, and the local maximum is the largest.

To find the local minimum and maximum of a function, you need to find the values of $x$ where the function's derivative is zero. Then, evaluate these points in the function's second derivative to determine if they are local minimum (when the second derivative is positive) or local maximum (when the second derivative is negative).

"No local minimum or maximum" means the function has no local minimum or maximum values. This can happen if there are no points where the derivative is zero or the second derivative is zero for all values that make the derivative zero.