1. The problem is to understand what a local minimum is in the context of a function. 2. A local minimum of a function $f(x)$ is a point $x = c$ where $f(c)$ is less than or equal to the values of $f(x)$ for all $x$ near $c$. 3. Mathematically, $f(c)$ is a local minimum if there exists an interval $(a,b)$ containing $c$ such that for all $x$ in $(a,b)$, $f(c) \leq f(x)$. 4. To find local minima, we often use the first and second derivative tests: - First derivative test: Find points where $f'(x) = 0$ or $f'(x)$ is undefined (critical points). - Second derivative test: If $f''(c) > 0$ at a critical point $c$, then $f(c)$ is a local minimum. 5. Example: For $f(x) = x^2$, - $f'(x) = 2x$, set $2x=0$ gives $x=0$. - $f''(x) = 2$, which is positive. - Therefore, $x=0$ is a local minimum with $f(0) = 0$. 6. In plain language, a local minimum is a point where the function dips down to a low point compared to nearby points, like the bottom of a valley. 7. This concept is important in optimization and calculus to find points where functions achieve minimum values locally.