Chapter 4.1, Problem 51E

(a)

To determine

To find : $f^{\prime }\left(2\right)$ does exist or not.

Answer to Problem 51E

  $f^{\prime }\left(2\right)$ does not exist.

Explanation of Solution

Given information :

  $f\left(x\right)={\left(x-2\right)}^{2/3}$

Formula needed :

For critical points, $\frac{dy}{dx}=0$ or $\frac{dy}{dx}$ is not defined.

Differentiate $f\left(x\right)={\left(x-2\right)}^{2/3}$ with respect to $x$ .

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}{\left(x-2\right)}^{2/3}\\ =\frac{2}{3}{\left(x-2\right)}^{2/3-1}\cdot \frac{d}{dx}\left(x-2\right)\\ =\frac{2}{3}{\left(x-2\right)}^{-1/3}\cdot \left(1\right)\\ =\frac{2}{3{\left(x-2\right)}^{1/3}}\end{array}$

As $x$ approach to 2 from left hand side, the value of function $f^{\prime }\left(x\right)$ tends to $-\infty$ .

As $x$ approach to 2 from right hand side, the value of function $f^{\prime }\left(x\right)$ tends to $\infty$ .

So, the derivative of function $f$ does not exist at $x=2$ .

(b)

To determine

To show : the only local extreme value of $f$ occur at $x=2$ .

Explanation of Solution

Given information :

  $f\left(x\right)={\left(x-2\right)}^{2/3}$

Formula needed :

For critical points, $\frac{dy}{dx}=0$ or $\frac{dy}{dx}$ is not defined.

The derivative of $f\left(x\right)={\left(x-2\right)}^{2/3}$ is $f^{\prime }\left(x\right)=\frac{2}{3{\left(x-2\right)}^{1/3}}$ .

Only one critical point exists at $x=2$ and no end point.

As $x$ approach to 2 from left hand side, the value of function $f\left(x\right)$ tends to 0.

As $x$ approach to 2 from right hand side, the value of function $f\left(x\right)$ tends to 0.

Hence, the only local extreme value of $f$ occur at $x=2$ .

(c)

To determine

To check : the result in part (b) and extreme value theorem.

Explanation of Solution

Given information :

  $f\left(x\right)={\left(x-2\right)}^{2/3}$

Formula needed :

For critical points, $\frac{dy}{dx}=0$ or $\frac{dy}{dx}$ is not defined.

According to extreme value theorem, if a function has a any extreme value at an interior point of its domain and derivative of function exists at that interior point, then the value of derivative function at that interior point is zero.

From part (b) the interior point is 2 but the derivative of function at 2 does not exist.

It contradicts the extreme value theorem.

(d)

To determine

To find : $f^{\prime }\left(a\right)$ does exist or not. Show that the only extreme value exists at $x=a$ .

Answer to Problem 51E

  $f^{\prime }\left(a\right)$ does not exist.

Explanation of Solution

Given information :

  $f\left(x\right)={\left(x-a\right)}^{2/3}$

Formula needed :

For critical points, $\frac{dy}{dx}=0$ or $\frac{dy}{dx}$ is not defined.

Differentiate $f\left(x\right)={\left(x-a\right)}^{2/3}$ with respect to $x$ .

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}{\left(x-a\right)}^{2/3}\\ =\frac{2}{3}{\left(x-a\right)}^{2/3-1}\cdot \frac{d}{dx}\left(x-a\right)\\ =\frac{2}{3}{\left(x-a\right)}^{-1/3}\cdot \left(1\right)\\ =\frac{2}{3{\left(x-a\right)}^{1/3}}\end{array}$

As $x$ approach to $a$ from left hand side, the value of function $f^{\prime }\left(x\right)$ tends to $-\infty$ .

As $x$ approach to $a$ from right hand side, the value of function $f^{\prime }\left(x\right)$ tends to $\infty$ .

So, the derivative of function $f$ does not exist at $x=a$ .

Only one critical point exists at $x=a$ and no end point.

As $x$ approach to $a$ from left hand side, the value of function $f\left(x\right)$ tends to 0.

As $x$ approach to $a$ from right hand side, the value of function $f\left(x\right)$ tends to 0.

Hence, the only local extreme value of $f$ occur at $x=a$ .