Question

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

(a) On what intervals \(f\) increasing? Explain.

(b) At what values of \(x\) does \(f\) have a local maximum or minimum? Explain.

(c) On what intervals is \(f\) concave upward or concave downward? Explain.

(d) What are the \(x\)-coordinates of the inflection points of \(f\)? Why?

Short answer

(a) Increasing on \(\left( {0,4} \right)\) and \(\left( {6,8} \right)\)

(b) Local maximum at \(x = 4,8\) and local minimum at \(x = 6\)

(c) Concave up on \(\left( {0,1} \right),\left( {2,3} \right)\) and \(\left( {5,7} \right)\). Concave down on \(\left( {1,2} \right)\), \(\left( {3,5} \right)\) and \(\left( {7,9} \right)\)

(d) Inflection points at \(x = 1,2,3,5,7\)

Step-by-step solution

(a) Step 1: Interval of increasing

A function is increasing in the interval where the derivative is positive.In the given figure, the graph of \(f'\left( x \right)\) is positive in the intervals \(\left( {0,4} \right)\) and \(\left( {6,8} \right)\).

Hence, the function in increasing on the intervals \(\left( {0,4} \right)\) and \(\left( {6,8} \right)\).

(b) Step 2: Local maximum and Local minimum

A function has local maximum and local minimum when the derivative changes from positive to negative and negative to positive respectively.

In the graph, \(f'\left( x \right)\) changes from positive to negative at \(x = 4\) and \(x = 8\). Also, \(f'\left( x \right)\) changes from negative to positive at \(x = 6\).

Hence, the function has local maximum at \(x = 4\) and \(x = 8\). Also, the function has local minimum at \(x = 6\).

(c) Step 3: Concavity

A function is concave upward when the derivative is increasing and concave down when the derivative is decreasing.

In the graph, \(f'\left( x \right)\) is increasing on intervals \(\left( {0,1} \right)\), \(\left( {2,3} \right)\) and \(\left( {5,7} \right)\). Also, \(f'\left( x \right)\) is decreasing on the intervals \(\left( {1,2} \right)\), \(\left( {3,5} \right)\) and \(\left( {7,9} \right)\).

Hence, \(f\) is concave up on the intervals \(\left( {0,1} \right)\), \(\left( {2,3} \right)\) and \(\left( {5,7} \right)\). Also, \(f\) is concave down on the intervals \(\left( {1,2} \right)\), \(\left( {3,5} \right)\) and \(\left( {7,9} \right)\).

(d) Step 4: Inflection points

The point where the concavity of the function changes is considered as the inflection point.

In the graph, the concavity changes at \(x = 1\), \(x = 2\), \(x = 3\), \(x = 5\) and \(x = 7\).

Hence, the \(x\)-coordinates of points of inflection are \(x = 1\), \(x = 2\), \(x = 3\), \(x = 5\) and \(x = 7\).