Chapter 1.5, Problem 4E

To determine

Fill the blanks

Answer to Problem 4E

A function value $f\left(a\right)$ is a relative -------- of $f$ if there exists an interval $\left(x_1,x_2\right)$ containing a such that $x_1<x<x_2$ implies $f\left(a\right)\ge f\left(x\right)$

Explanation of Solution

Given:

A function value $f\left(a\right)$ is a relative maximum of $f$ if there exists an interval $\left(x_1,x_2\right)$ containing a such that $x_1<x<x_2$ implies $f\left(a\right)\ge f\left(x\right)$

A function value $f\left(a\right)$ is a relative maximum of $f$ if there exists an interval $\left(x_1,x_2\right)$ containing a such that $x_1<x<x_2$ implies $f\left(a\right)\ge f\left(x\right)$

Consider an interval $\left(x_1,x_2\right)$

Let p belongs to interval $\left(x_1,x_2\right)$ such that $x_1<x<x_2$

If a function $f\left(p\right)$ is a relative minimum, then $f\left(p\right)\le f\left(x\right)$

If a function

  $f\left(p\right)$ is a relative maximum, then $f\left(p\right)\ge f\left(x\right)$