Chapter 1.3, Problem 22E

Solution

To Find: The suitable functions from twelve basic functions based on the given description.

The three functions with infinitely many local extrema are:

  $\begin{array}{c}y=\ln x\\ y=\sin x\\ y=\cos x\end{array}$

Given information:

The three functions with infinitely many local extrema.

Given functions are:

  $\begin{array}{l}1.f\left(x\right)=x2.f\left(x\right)=x^2\\ 3.f\left(x\right)=x^{3}4.f\left(x\right)=\frac{1}{x}\\ 5.f\left(x\right)=\sqrt{x}6.f\left(x\right)=e^x\\ 7.f\left(x\right)=\ln x8.f\left(x\right)=\sin x\\ 9.f\left(x\right)=\cos x10.f\left(x\right)=\left|x\right|\\ 11.f\left(x\right)=\mathrm{int}x12.f\left(x\right)=\frac{1}{1+e^{-x}}\end{array}$

Calculation:

An open interval containing $c$ is said to have a local maximum if the value of $f\left(c\right)$ exceeds or equals all range values of $f$ .

If a value $f\left(c\right)$ of a function $f$ is less than or equal to all range values of $f$ on some interval including $c$ , then it is called the local minimum of the function.

From the given functions, the three functions with infinitely many local extrema. are:

  $y=\ln x$

  

$y=\sin x$

  

  $y=\cos x$

  

Thus, the three functions with infinitely many local extrema are:

  $\begin{array}{c}y=\ln x\\ y=\sin x\\ y=\cos x\end{array}$