Question

The figure shows the graph of the derivative\(f'\)of a function\(f\).

(a) On what intervals\(f\)is increasing or decreasing?

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

(c) Sketch the graph of\(f''\).

(d) Sketch a possible graph of\(f\).

Short answer

(a)\(f\)is increasing on the intervals \(\left( { - 2,0} \right)\), \(\left( {4,\infty } \right)\) and decreasing on the interval \(\left( { - \infty , - 2} \right)\), \(\left( {0,4} \right)\).

(b) \(x = - 2, - 4\)is the local minimum and\(x = 0\)is the local maximum.

(c) The graph of\(f''\)is drawn.

(d) The graph of \(f\) is drawn.

Step-by-step solution

Intervals of \(f\left( x \right)\) increasing or decreasing

A function \(f\) is increasing if \(f'\left( x \right) > 0\) and decreasing \(f'\left( x \right) < 0\).

From the graph, \(f\) is increasing on the intervals \(\left( { - 2,0} \right)\), \(\left( {4,\infty } \right)\) and decreasing on the interval \(\left( { - \infty , - 2} \right)\), \(\left( {0,4} \right)\).

local maximum and minimum of \(f\)

If\(f'\left( c \right) = 0\)and\(f'\)changes sign from negative to positive at c, then\(f\left( c \right)\)is the local minimum.

From the graph, the sign of\(f'\)changes from negative to positive at\(x = - 2, - 4\).

Thus,\(x = - 2, - 4\)is the local minimum.

If\(f'\left( c \right) = 0\)and\(f'\)changes sign from positive to negative at c, then\(f\left( c \right)\)is the local maximum.

From the graph, the sign of\(f'\)changes from positive to negative at\(x = 0\).

Thus, \(x = 0\) is the local maximum.

The graph of \(f''\)

Draw the graph of the function \(f''\) as shown below:

The graph of \(f\)

Draw the graph of the function \(f\) as shown below:

Therefore, the graph of the given function is drawn.