Question

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

37. \(f'\left( {\bf{0}} \right) = f'\left( {\bf{2}} \right) = f'\left( {\bf{4}} \right) = {\bf{0}}\), \(f'\left( x \right) > {\bf{0}}\)if \(x < {\bf{0}}\) or \({\bf{2}} < x < {\bf{4}}\),

\(f'\left( x \right) < {\bf{0}}\)if \({\bf{0}} < x < {\bf{2}}\) or \(x > {\bf{4}}\),

\(f''\left( x \right) > {\bf{0}}\)if \[{\bf{1}} < x < {\bf{3}}\], \(f''\left( x \right) < {\bf{0}}\) if \[x < {\bf{1}}\] or \[x > {\bf{3}}\].

Short answer

The required graph is given below.

Step-by-step solution

Write an interpretation of the given information about  f

The following observations can be made about f from the given information:

1. The equation\(f'\left( 0 \right) = f'\left( 2 \right) = f'\left( 4 \right) = 0\)shows that there are horizontal tangents at\(x = 0\),\(x = 2\), and\(x = 4\).
2. As\(f'\left( x \right) > 0\)and\(x < 0\)or\(2 < x < 4\), it shows that f is increasing in the intervals\(\left( { - \infty ,0} \right)\)and\(\left( {2,4} \right)\).
3. As\(f'\left( x \right) < 0\)and\(0 < x < 2\)or\(x > 4\), it shows that f is decreasing in the intervals\[\left( {0,2} \right)\]and\(\left( {4,\infty } \right)\).
4. As\(f''\left( x \right) > 0\)for\(1 < x < 3\), f is concave upward in\(\left( {1,3} \right)\).
5. As\(f''\left( x \right) < 0\)for\(x < 1\)or\(x > 3\), f is concave downward in \(\left( { - \infty ,1} \right)\).

Sketch the possible graph of \(f\left( x \right)\)

By using the nature of the curve given in Step 1, plot the sketch of the curve of\(f\left( x \right)\).

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