Chapter 5.1, Problem 51E

a.

To determine

To state: that $f\text{'}\left(2\right)$ exist or not.

Answer to Problem 51E

the $f\text{'}\left(2\right)$ does not exist.

Explanation of Solution

Given information:

The function is $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ .

since, in the function $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ , the derivative of the function is

  $f\text{'}\left(x\right)=\frac{2}{3{\left(x-2\right)}^{\frac{1}{3}}}$

Thus, the $f\text{'}\left(2\right)$ does not exist.

b.

To determine

To show: that the only local extreme value of function occurs at $x=2$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ .

Proof :since, in the function $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ , the derivative of the function is

  $f\text{'}\left(x\right)=\frac{2}{3{\left(x-2\right)}^{\frac{1}{3}}}$

Thus, the derivative is defined and nonzero for $x\ne 2$ , and $f\left(2\right)=0$ and $f\left(x\right)>0$ for all $x\ne 2$ .

c.

To determine

To state: that the result in (b) contradict the extreme value theorem or not.

Answer to Problem 51E

The result in (b) does not contradict the extreme value theorem.

Explanation of Solution

Given information:

The function is $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ .

since, in the function $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ , the derivative of the function is

  $f\text{'}\left(x\right)=\frac{2}{3{\left(x-2\right)}^{\frac{1}{3}}}$

The result in (b) does not contradict the extreme value theorem, because the domain of the function is $\left(-\infty ,\infty \right)$ and the domain is not a close interval.

d.

To determine

To state: that $f\text{'}\left(2\right)$ exist or not and show that the only local extreme value of function occurs at $x=2$ .

Answer to Problem 51E

the $f\text{'}\left(2\right)$ does not exist.

Explanation of Solution

Given information:

The function is $f\left(x\right)={\left(x-a\right)}^{\frac{2}{3}}$ .

replace the $a\text{ by 2}$ ,

The function is $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ .

since, in the function $f\left(x\right)={\left(x-2\right)}^{\frac{2}{3}}$ , the derivative of the function is

  $f\text{'}\left(x\right)=\frac{2}{3{\left(x-2\right)}^{\frac{1}{3}}}$

Thus, the $f\text{'}\left(2\right)$ does not exist.

Thus, the derivative is defined and nonzero for $x\ne 2$ , and $f\left(2\right)=0$ and $f\left(x\right)>0$ for all $x\ne 2$ .