Question

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

(a) On what intervals is \(f\) increasing? Decreasing?

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

6.

Short answer

(a) Increasing on \(\left( {1,4} \right)\), \(\left( {5,6} \right)\) and decreasing on \(\left( {0,1} \right)\), \(\left( {4,5} \right)\).

(b) Local maxima at \(x = 4\) and local minima at \(x = 1\), \(x = 5\).

Step-by-step solution

The first and second derivative tests

The first and second derivative tests are utilized to find the point of local maximum or minimum of the function.

Intervals of increasing or decreasing

(a)

In the figure, it can be seen that the graph of the given function is less than 0 in the intervals \(\left( {0,1} \right)\), \(\left( {4,5} \right)\). So, the function is decreasing on \(\left( {0,1} \right)\), \(\left( {4,5} \right)\).

Also, the graph of the given function is greater than 0 in the intervals \(\left( {1,4} \right)\), \(\left( {5,6} \right)\). So, the function is increasing on \(\left( {1,4} \right)\), \(\left( {5,6} \right)\).

Hence, the function increasing on \(\left( {1,4} \right)\), \(\left( {5,6} \right)\) and decreasing on \(\left( {0,1} \right)\), \(\left( {4,5} \right)\)

Local minima and local maxima

(b)

At \(x = 4\), the value of \(f'\left( x \right) = 0\) and \(f'\left( x \right)\) changes from positive to negative at these values. Therefore, \(f\) has a local maximum at \(x = 4\).

At \(x = 1\) and \(x = 5\), the value of \(f'\left( x \right) = 0\) and \(f'\left( x \right)\) changes its sign from negative to positive. Therefore, \(f\) has a local minimum at \(x = 1\) and \(x = 5\).

Hence, the function has local maxima at \(x = 4\) and local minima at \(x = 1\) and \(x = 5\).