Chapter 5.3, Problem 40PPS

(a)

To determine

To find:

The graph of $f$ has a vertical tangent or a vertical cusp at c .

Explanation of Solution

Given:

The graph of the function $f\left(x\right)$

Concept used:

The definition of a vertical cusp is that the one sided limits of the derivative approach opposite $\pm \infty$ : positive infinity on one side and negative infinity on the other side . A vertical tangent has the one sided limits of the derivative equal to the same sign of infinity

The derivative at the relevant point is undefined in both the cusp and the vertical tangent

Calculation:

As the $x$ -value approach negative infinity , the $y$ - value approach positive infinity

As the $x$ -value approach positive infinity , the $y$ - value approach negative infinity

  $\begin{array}{l}f\left(x\right)\to -\infty asx\to -\infty \\ f\left(x\right)\to -\infty asx\to \infty \end{array}$

The end behavior is in same directions.

(b)

To determine

To find:

The graph of $f$ has a vertical tangent or a vertical cusp at c .

Explanation of Solution

Given:

The graph of the function $f\left(x\right)$

Concept used:

The definition of a vertical cusp is that the one sided limits of the derivative approach opposite $\pm \infty$ : positive infinity on one side and negative infinity on the other side . A vertical tangent has the one sided limits of the derivative equal to the same sign of infinity

The derivative at the relevant point is undefined in both the cusp and the vertical tangent

Calculation:

As the $x$ -value approach negative infinity , the $y$ - value approach positive infinity

As the $x$ -value approach positive infinity , the $y$ - value approach negative infinity

  $\begin{array}{l}f\left(x\right)\to -\infty asx\to -\infty \\ f\left(x\right)\to -\infty asx\to \infty \end{array}$

The end behavior is in same directions.

Since the end behavior is in same directions , and is an even − degree function.

(c)

To determine

To find:

The graph of $f$ has a vertical tangent or a vertical cusp at c .

Explanation of Solution

Given:

The graph of the function $f\left(x\right)$ .

Concept used:

The definition of a vertical cusp is that the one sided limits of the derivative approach opposite $\pm \infty$ : positive infinity on one side and negative infinity on the other side . A vertical tangent has the one sided limits of the derivative equal to the same sign of infinity

The derivative at the relevant point is undefined in both the cusp and the vertical tangent

Calculation:

As the $x$ -value approach negative infinity , the $y$ - value approach positive infinity

As the $x$ -value approach positive infinity , the $y$ - value approach negative infinity

  $\begin{array}{l}f\left(x\right)\to -\infty asx\to -\infty \\ f\left(x\right)\to -\infty asx\to \infty \end{array}$

The end behavior is in same directions.

Since the end behavior is in same directions , and is an even − degree function.

The graph intersects the $x$ -axis at zero points ,

So there are two real zeros .