Local extrema exists at a point where the derivative of a function changes sign. The function has a local MAXIMUM at a critical point of the derivative changes from positive to negative. The function has a local MINIMUM if the derivative changes from negative to positive. The First Derivative Test for local extrema is used to determine if the function has a local maximum or minumum value at that crtitical point.
Here we use that test to find the local extrema for the function: $f\left(x\right)={x}^{4}-8{x}^{2}$
First Derivative Test and Local Extrema