Question

1. Find the intervals of increase or decrease.
2. Find the local maximum and minimum values.
3. Find the intervals of concavity and the inflection points.
4. Use the information from parts (a)-(c) to sketch the graph. You may want to check your work with a graphing calculator or computer.

45. \(f\left( x \right) = {x^3} - 3{x^2} + 4\)

Short answer

a) The function \(f\) is increasing at \(\left( {2,\infty } \right)\). The function \(f\) is decreasing at \(\left( {0,2} \right)\).

b) The local maximum value is \(f\left( 0 \right) = 4\) , and the local minimum value is \(f\left( 2 \right) = 0\).

c)The function \(f\) is concave upward on \(\left( {1,\infty } \right)\) and \(f\) is concave downward on \(\left( { - \infty ,1} \right)\). The point \(\left( {1,2} \right)\) is an inflection point.

d) The graph of \(f\) as shown below:

Step-by-step solution

Increasing/ Decreasing Test, concavity Test

Theincreasing and decreasing testas shown below:

1. The function \(f\) isincreasing on the interval when \(f'\left( x \right) > 0\) on an interval.
2. The function \(f\) is decreasingon the interval when \(f'\left( x \right) < 0\) on an interval.

Determine the intervals of increase or decrease

a)

Obtain the derivative of \(f\) as shown below:

\(\begin{array}{c}f'\left( x \right) = 3{x^2} - 6x\\ = 3x\left( {x - 2} \right)\end{array}\)

Take \(f'\left( x \right) = 0\) to obtain the critical numbers as shown below:

\(\begin{array}{c}3x\left( {x - 2} \right) = 0\\x\left( {x - 2} \right) = 0\\x = 0,2\end{array}\)

The critical number are \(x = 0\) and \(x = 2\).

Interval	\(3x\)	\(x - 2\)	\(f'\left( x \right)\)	\(f\)
\(x < 0\) \(0 < x < 2\) \(x > 2\)	\( - \) \( + \) \( + \)	\( - \) \( - \) \( + \)	\( + \) \( - \) \( + \)	Increasing on \(\left( { - \infty ,0} \right)\) Decreasing on \(\left( {0,2} \right)\) Increasing on \(\left( {2,\infty } \right)\)

Thus, the function \(f\) is increasing at \(\left( {2,\infty } \right)\). The function \(f\) is decreasing at \(\left( {0,2} \right)\).

Determine the local maximum and local minimum values

b)

Thefirst derivative test: Let \(c\) be the critical number of a continuous function \(f\).

1. The function \(f\) has a local maximum at \(c\) when \(f'\) changes from positive to negative at \(c\).
2. The function \(f\) has a local minimum at \(c\) when \(f'\) changes from negative to positive at \(c\).
3. When \(f'\) is positive to the left and right of \(c\), or negative to the left and right of \(c\), then \(f\) contain no local maximum or minimum at \(c\).

There are changes in \(f\) from increasing to decreasing at \(x = 0\) and from decreasing to increasing at \(x = 2\).

Thus, the local maximum value is \(f\left( 0 \right) = 4\) and the local minimum value is \(f\left( 2 \right) = 0\).

Determine the interval of concavity and the inflection points

c.

The concavity test as shown below:

1. When \(f''\left( x \right) > 0\) on an interval, \(I\)then the graph of \(f\) is said to be concave upward on \(I\).
2. When \(f''\left( x \right) < 0\) on an interval, \(I\)then the graph of \(f\) is said to be concave downward on \(I\).

Obtain the second derivative of the function as shown below:

\(\begin{array}{c}f''\left( x \right) = 6x - 6\\ = 6\left( {x - 1} \right)\end{array}\)

Take \(f''\left( x \right) = 0\) to obtain the critical number as shown below:

\(\begin{array}{c}6\left( {x - 1} \right) = 0\\x - 1 = 0\\x = 1\end{array}\)

The critical point of \(f\) is \(x = 1\). \(f''\left( x \right) > 0\) on the interval \(\left( {1,\infty } \right)\) and \(f''\left( x \right) < 0\) on the interval \(\left( { - \infty ,1} \right)\).

Thus, the function \(f\) is concave upward on \(\left( {1,\infty } \right)\) and \(f\) is concave downward on \(\left( { - \infty ,1} \right)\). The point \(\left( {1,2} \right)\) is an inflection point.

Sketch the graph of f

e)

The function \(f\) is increasing at \(\left( {2,\infty } \right)\). The function \(f\) is decreasing at \(\left( {0,2} \right)\).The point \(\left( {1,2} \right)\) is an inflection point. The local maximum value is \(f\left( 0 \right) = 4\) and the local minimum value is \(f\left( 2 \right) = 0\).

Use the above condition to sketch the graph of \(f\) as shown below:

Thus, the graph of the function \(f\) is obtained.