Question

What are local extrema?

Answer 1

Points on some function where a local maximum or minimum value occurs. For a continuous function over its entire domain, these points exist where the slope of the function `=0` (i.e it's first derivative is equal to 0).

Explanation:

Consider some continuous function `f(x)`

The slope of `f(x)` is equal to zero where `f'(x)=0` at some point `(a, f(a))`. Then `f(a)` will be a local extreme value (maximim or minimum) of `f(x)`

N.B. Absolute extrema are a subset of local extrema. These are the points where `f(a)` is the extreme value of `f(x)` over its entire domain.