Q. 1 - Calculus

Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• $f$ has a local maximum or a local minimum at $x=c$ .

Step-by-Step Solution

Answer

Ans: part (a). At local maximum $x=c$, if there exists some $\delta > 0$ such that $f \left(c\right) \ge f \left(x\right)$ for all $x \in (c - \delta , c + \delta )$ .

part (b). At local minimum $x=c$ , if there exists some $\delta > 0$ such that $f \left(c\right) \le f \left(x\right)$ for all $x \in (c - \delta , c + \delta ).$

Step 1. Given Information:

Function $f$ has a local maximum or a local minimum at

Step 2. Graphical representation:

Intuitively, at a local extremum, the tangent line of a function must be either horizontal or

undefined; for example, consider the graphs: