Q. 3 - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

1. The graph of a function f for which f' is positive everywhere, $f^{\text{'}\text{'}}\left(x\right)>0$ for $x<-2$, and $f^{\text{'}\text{'}}\left(x\right)<0$ for $x>-2$.
2. The graph of a function f for which $f\left(3\right)=0$, $f^{\text{'}}\left(3\right)=0$, and $f^{\text{'}\text{'}}\left(3\right)=0$.
3. The graph of a function f for which f(x) is zero at $x=-1, x=2$, and $x=4;f^{\text{'}}\left(x\right)$ is zero at $x=-1, x=1$, and $x=3$; and f'' is zero at $x=0$ and $x=2$.

Step-by-Step Solution

Answer

Part(a) The example is $f\left(x\right)=-(x+2{)}^{-5}$.

Part(b) The example is$f\left(x\right)={(x-3)}^5$

Part(c) It is not given which interval gives a positive result or which interval gives a negative result. Hence it is not possible to sketch such a function.

Part(a) Step 1. Given Information.

We are given a function f(x) and different conditions.

a. The graph of a function $f$ for which $f^{\text{'}}$ is positive everywhere,$f^{\text{'}\text{'}}\left(x\right)>0$ for $x<-2$, and $f\text{'}\left(x\right)<0$for$x>-2$.

b. The graph of a function $f$ for which $f\left(3\right)=0$, $f^{\text{'}}\left(3\right)=0$, and $f^{\text{'}\text{'}}\left(3\right)=0$.

c. The graph of a function $f$ for which $f\left(x\right)$ is zero at $x=-1,x=2$, and $x=4, f^{\text{'}}\left(x\right)$ is zero at $x=-1,x=1$, and $x=3$, and $f^{\text{'}\text{'}}$ is zero at $x=0$ and $x=2.$

Part(a) Step 1. Finding the example

The graph of a function f for which $f^{\text{'}}$ is positive everywhere, $f^{\text{'}\text{'}}\left(x\right)>0$ for $x<-2$, and $f^{\text{'}\text{'}}\left(x\right)<0$ for $x>-2$,

Let the function be,

$f\left(x\right)=-(x+2{)}^{-5}$

The graph is as follows,

Part(a) Step 1. Checking the derivative

The derivative is given by,

$$\begin{gathered} f^{\text{'}}\left(x\right)=-(-5)(x+2{)}^{-5-1} \\ =5(x+2{)}^{-6} \\ =\frac{5}{(x+2{)}^6} \end{gathered}$$

And,

$$\begin{gathered} f^{\text{'}\text{'}}\left(x\right)=\frac{d}{dx}5(x+2{)}^{-6} \\ =5(-6\left)\right(x+2{)}^{-6-1} \\ =-30(x+2{)}^{-7} \end{gathered}$$

Hence, the example is $f\left(x\right)=-(x+2{)}^{-5}$ .

Part (b) Step 1. Finding example

The graph of a function $f$ for which$f\left(3\right)=0,f^{\mathrm{\prime }}\left(3\right)=0$ and$f^{\prime \prime }\left(3\right)=0$

Let the function be $f\left(x\right)=(x-3{)}^5$

The graph is as follows,

Part(b) Step 2. Checking the derivative

The derivative is given by,

$\begin{array}{rl}{f}^{\mathrm{\prime }}\left(x\right)& =5(x-3{)}^{5-1}\\ & =5(x-3{)}^4\end{array}$

And

$\begin{array}{rl}{f}^{\prime \prime }\left(x\right)& =5\left(4\right(x-3{)}^{4-1})\\ & =20(x-3{)}^3\end{array}$

Hence the function is$f\left(x\right)=(x-3{)}^5$

Part (c) Step 1. Finding example

The graph of a function $f$ for which $f\left(x\right)$ is zero at $x=-1,x=2$ and $x=4, f^{\text{'}}\left(x\right)$ is zero at $x=-1,x=1$ and $x=3$; and $f^{\text{'}\text{'}}\left(x\right)$ is zero at $x=0$ and $x=2$.

Here, it is not given that which interval gives a positive result or which interval gives a negative result. Hence it is not possible to sketch such a function.