Question

The graph of the derivative \(f'\) of a continuous function \(f\) is shown

1. On what intervals is f increasing? Decreasing?
2. At what values of \(x\) does f have a local maximum? Local minimum?
3. On what intervals is f concave upward? Concave downward?
4. State the \(x\)-coordinate(s) of the point(s) of inflection.
5. Assuming that \(f\left( 0 \right) = 0\), sketch the graph of f.

43.

Short answer

1. The function\(f\)is increasing on the interval\(\left( {0,2} \right),\left( {4,6} \right),\)and \(\left( {8,\infty } \right)\).The function\(f\)is decreasing on the interval\(\left( {2,4} \right)\)and \(\left( {6,8} \right)\).
2. The function \(f\)contains local maxima at \(x = 2\) and \(x = 6\). \(f\) has local minima at \(x = 4\) and \(x = 8\).
3. The function \(f\) is concave upward on the interval \(\left( {3,6} \right)\) , and \(\left( {6,\infty } \right)\). \(f\) is concave downward on the interval \(\left( {0,3} \right)\).
4. The point of inflection occurs at \(x = 3\).
5. The graph of \(f\) as shown below:

Step-by-step solution

Increasing/ Decreasing Test, concavity Test

Theincreasing and decreasing test as shown below:

1. The function \(f\) is increasing 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.

TheConcavity 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 \(I\).

Determine at what interval is \(f\) increasing or decreasing

a)

It is observed from the graph that the function \(f\) is increasing when \(f'\) is positive, therefore on the interval \(\left( {0,2} \right),\left( {4,6} \right),\) and \(\left( {8,\infty } \right)\).

It is observed from the graph that function \(f\) is decreasing when \(f'\) is negative, therefore on the interval \(\left( {2,4} \right)\) and \(\left( {6,8} \right)\).

Determine the values of x does \(f\) have a local maximum and local minimum

b)

Thesecond derivative test: Let \(f''\) be continuous near \(c\).

1. When \(f'\left( c \right) = 0\) and \(f''\left( c \right) > 0\) then function \(f\) containlocal minimum at \(c\).
2. When \(f'\left( c \right) = 0\) and \(f''\left( c \right) < 0\) then function \(f\) contains local maximum at \(c\).

The graph shows that there are changes in \(f'\) from positive to negative at \(x = 2\) and \(x = 6\) such that the function \(f\)contains local maxima.

There are changes in \(f'\) from negative to positive at \(x = 4\) and \(x = 8\) such that the function \(f\) has local minima.

Determine at what interval is \(f\) concave upward and concave downward

c)

It is observed from the graph that the function \(f\) is concave upward (CU) when \(f'\) is increasing, therefore on the interval \(\left( {3,6} \right)\) and \(\left( {6,\infty } \right)\).

The function \(f\) is concave downward (CD) when \(f'\) is decreasing, therefore on the interval \(\left( {0,3} \right)\).

State the x-coordinate(s) of the point(s) of inflection

d)

The point of inflection occurs at \(x = 3\) because the function \(f\) changes from concave downward to concave upward.

Sketch the graph of f

e)

The point of inflection occurs at \(x = 3\). \(f\) is concave upward at \(\left( {3,6} \right)\),\(\left( {6,\infty } \right)\) and concave downward at \(\left( {0,3} \right)\).

Consider that \(f\left( 0 \right) = 0\). This leads to the point \(\left( {0,0} \right)\).

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