1. The problem is to understand what a local extremum is in mathematics. 2. A local extremum refers to a point on a function's graph where the function reaches a local maximum or minimum value. 3. A local maximum is a point where the function's value is greater than all nearby points. 4. A local minimum is a point where the function's value is less than all nearby points. 5. Formally, a function $f(x)$ has a local maximum at $x=a$ if there exists an interval around $a$ such that for all $x$ in that interval, $f(a) \geq f(x)$. 6. Similarly, $f(x)$ has a local minimum at $x=b$ if there exists an interval around $b$ such that for all $x$ in that interval, $f(b) \leq f(x)$. 7. Local extrema are important in calculus and optimization because they indicate points where the function changes direction. 8. These points can be found by setting the derivative $f'(x)$ equal to zero and analyzing the sign changes of $f'(x)$ around those points.