Question

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f'\) and \(f''\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

3. \(f\left( x \right) = {x^6} - 5{x^5} + 25{x^3} - 6{x^2} - 48x\)

Short answer

The obtained Graph for the functions \(f,{\rm{ }}f',{\rm{ and }}f''\) are shown below:

Step-by-step solution

Analysing a Graph for any Function:

There are the following terms that can be examined using theGraphof a functionand that ofits derivatives:

1. The Domainof the Function.
2. Intervals ofIncrease and Decrease.
3. TheMaxima and Minimaof that function.
4. The Concavity and the Point of Inflection.

Solving for Domain.

The given function is:

\(f\left( x \right) = {x^6} - 5{x^5} + 25{x^3} - 6{x^2} - 48x\)

The steps to plot the Graph of given functions by using the graphing calculator are:

1. In the graphing calculator, select “STAT PLOT” and enter the equations \({x^6} - 5{x^5} + 25{x^3} - 6{x^2} - 48x\) in the \({Y_1}\) tab.
2. Set the window size \( - 2 \le X \le 4\), and \( - 40 \le Y \le 50\).
3. Enter the graph button in the graphing calculator.

Here, the Graph obtained using the graphing calculator is shown as:

The domain can be seen as:

\(D = \left\{ {x\left| {x \in \mathbb{R}} \right.} \right\}\)

So, the function exists for all real numbers.

Graphing for intervals of Increase and Decrease.

Now, for intervals of increase and decrease, differentiating \(f\left( x \right)\)as:

\(\begin{array}{c}f'\left( x \right) = \frac{d}{{dx}}\left( {{x^6} - 5{x^5} + 25{x^3} - 6{x^2} - 48x} \right)\\ = 6{x^5} - 25{x^4} + 75{x^2} - 12x - 48\end{array}\)

The steps to plot the Graph of given functions by using the graphing calculator are:

1. In the graphing calculator, select “STAT PLOT” and enter the equations \(6{x^5} - 25{x^4} + 75{x^2} - 12x - 48\) in the \({Y_1}\) tab.
2. Set the window size \( - 2 \le X \le 4\), and \( - 60 \le Y \le 50\).
3. Enter the graph button in the graphing calculator.

On graphing \(f'\left( x \right)\)with the help of graphing calculator, we get:

From the Graph, we see:

\(f'\left( x \right) < 0\)

And, \(f\)is decreasing on \(\left( { - \infty , - 1.31} \right){\rm{,}}\,\,\left( { - 0.84,1.06} \right){\rm{ and }}\left( {2.50,2.75} \right)\).

Also,

\(f'\left( x \right) > 0\)

And, \(f\)is increasing on \(\left( { - 1.31, - 0.84} \right){\rm{,}}\left( {1.06,2.50} \right){\rm{ and }}\left( {2.75,\infty } \right)\).

Thus, the local minimum values will be:

\(\begin{array}{l}f\left( { - 1.31} \right) \approx 20.72\\f\left( {1.06} \right) \approx - 33.12\\f\left( {2.75} \right) \approx - 11.33\end{array}\)

And, the local maximum values will be:

\(f\left( { - 0.84} \right) \approx 23.71{\rm{ and }}f\left( {2.50} \right) \approx - 11.02\)

Graphing for Concavity and Inflection points.

The concavity of the function can be examined using a Graph of the second derivative of the function. So, differentiating the derivative once again, we get:

\(\begin{array}{c}f''\left( x \right) = \frac{d}{{dx}}\left( {6{x^5} - 25{x^4} + 75{x^2} - 12x - 48} \right)\\ = 30{x^4} - 100{x^3} + 150x - 12\end{array}\)

The steps to plot the Graph of given functions by using the graphing calculator are:

1. In the graphing calculator, select “STAT PLOT” and enter the equations \(30{x^4} - 100{x^3} + 150x - 12\) in the \({Y_1}\) tab.
2. Set the window size \( - 2 \le X \le 4\), and \( - 60 \le Y \le 50\).
3. Enter the graph button in the graphing calculator.

On graphing \(f''\left( x \right)\)with the help of a graphing calculator, we get:

From the Graph, we see:

\(f''\left( x \right) < 0\)

And \(f\)is concave downon \(\left( { - 1.10,0.08} \right){\rm{ and }}\left( {1.72,2.64} \right)\).

Also,

\(f''\left( x \right) > 0\)

And, \(f\)is concave upon \(\left( { - \infty , - 1.10} \right),\left( {0.08,1.72} \right){\rm{ and }}\left( {2.64,\infty } \right)\).

Thus, the Inflection Points will be at:

\(\left( { - 1.10,22.09} \right),\left( {0.08, - 3.88} \right),\left( {1.72, - 22.53} \right){\rm{ and }}\left( {2.64, - 11.18} \right)\)

Hence, these are the required answer.