Question

Sketch the graph of the function that satisfies all of the given conditions.

34. (a) \(f'\left( x \right) < {\bf{0}}\) and \(f''\left( x \right) < {\bf{0}}\), for all \(x\)

(b) \(f'\left( x \right) > {\bf{0}}\) and \(f''\left( x \right) > {\bf{0}}\), for all \(x\).

Short answer

The required graphs are given below.

(a)

(b)

Step-by-step solution

Find the answer for part (a)

As \(f'\left( x \right) < 0\), it shows that the function is always decreasing. Similarly, if \(f''\left( x \right) < 0\), it shows that it is concave downward.

The figure given below represents the possible curve of \(f\left( x \right)\).

Find the answer for part (b)

As \(f'\left( x \right) > 0\), it shows that the function is always increasing. Similarly, if \(f''\left( x \right) > 0\), it shows that it is concave upward.

The figure given below represents the possible curve of \(f\left( x \right)\).