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 $x=c$.

Short 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-by-step solution

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: