Question

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

39. \(f'\left( {\bf{5}} \right) = {\bf{0}}\), \(f'\left( x \right) < {\bf{0}}\)when \(x < {\bf{5}}\), \(f'\left( x \right) > {\bf{0}}\) when \(x > {\bf{5}}\), \(f''\left( {\bf{2}} \right) = {\bf{0}}\), \(f''\left( {\bf{8}} \right) = {\bf{0}}\)

\(f''\left( x \right) < {\bf{0}}\)if \[x < {\bf{2}}\] or \[x > {\bf{8}}\], \(f''\left( x \right) > {\bf{0}}\) if \[{\bf{2}} < x < {\bf{8}}\], \(\mathop {{\bf{lim}}}\limits_{x \to \infty } \,\,f\left( x \right) = {\bf{3}}\), \(\mathop {{\bf{lim}}}\limits_{x \to - \infty } \,\,f\left( x \right) = {\bf{3}}\)

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\(f'\left( 5 \right) = 0\)shows that the curve has a horizontal tangent at\(x = 5\).
2. As\(f'\left( x \right) < 0\)for\(x < 5\), f is decreasing in the interval\(\left( { - \infty ,5} \right)\).
3. As\(f'\left( x \right) > 0\)for\(x > 5\), f is increasing in the interval\(\left( {5,\infty } \right)\).
4. As\(f''\left( 2 \right) = 0\),\(f''\left( 8 \right) = 0\),\(f''\left( x \right) < 0\)when\(x < 2\)or\(x > 8\), f is concave upward in\(\left( { - \infty ,1} \right)\)and\(\left( {3,\infty } \right)\).
5. As\(f''\left( x \right) > 0\)for\(2 < x < 8\), f is concave upward in\(\left( {2,8} \right)\).
6. As\(f''\left( x \right) < 0\)for\(x < 2\)or\(x > 8\), f is concave downward in\(\left( { - \infty ,2} \right)\)and\(\left( {8,\infty } \right)\).
7. The expression \(\mathop {\lim }\limits_{x \to - \infty } f\left( x \right) = 3\) shows that \(y = 3\) is a horizontal asymptote.

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 \(f\left( x \right)\).