Question

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

(a) On what intervals is \(f\) increasing? Explain.

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

(c) On what intervals is \(f\) concave upward or concave downward? Explain.

(d) What are the \(x\)-coordinates of the inflection points of \(f\)? Why?

Short answer

(a) The function \(f\) is increasing over the intervals \((2,4)\) and \((6,9)\) and \(f\) is decreasing over the intervals \((0,2)\) and \((4,6)\).

(b) The local maximum is \((4,0)\) and the local minimum is \((2,0)\) and \((6,0)\).

(c) The function \(f\) is concave upward in \((1,3),(5,7)\) and \((8,9)\) and \(f\) is concave downward in \((0,1),(3,5)\) and \((7,8)\).

(d) The \(x\)-coordinate of the inflection point is \(x = 1,\;3,\;5,\;7

Step-by-step solution

Given data

The given curve is the graph of \(f\).

Concept of inflection point and concavity test

A point\(P\)on a curve under the function\(f(x)\)is called an inflection point if\(f\)is continuous and curve changes from concave upward to concave downward or from concave downward to concave upward at\(P\).

Concavity test:

(a) If\({f^{\prime \prime }}(x) > 0\)for all\(x\)in\(l\), then\(f\)is concave upward on\(l\).

(b) If\({f^{\prime \prime }}(x) < 0\)for all\(x\)in\(l\), then\(f\)is concave downward on\(l\).

(a) Step 3: Determine the intervals on which \(f\) is increasing

It is known that, \(f(x)\) is increasing over intervals for which \({f^\prime }(x) > 0\) and decreasing over intervals for which \({f^\prime }(x) < 0\).

From the given graph of \({f^\prime }(x)\), it is observed that \({f^\prime }(x) > 0\) for the intervals \((2,4)\) and \((6,9)\) and \({f^\prime }(x) < 0\) for the intervals \((0,2)\) and \((4,6)\).

Thus, \(f\) is increasing over the intervals \((2,4)\) and \((6,9)\) and \(f\) is decreasing over the intervals \((0,2)\) and \((4,6)\).

(b) Step 4: Determine the values of \(x\) at which \(f\) has local maximum or minimum

It is known that, if \({f^\prime }(x)\) changes from negative to positive at a critical number \(c\) then \(f(c)\) is a local maximum.

Similarly, if \({f^\prime }(x)\) changes from positive to negative at a critical number \(c\) then \(f(c)\) is a local minimum.

From the given graph of \({f^\prime }(x)\), it is observed that at \(x = 2\) and \(x = 6\) the curve changes from negative to positive.

Thus, the local minimum is \((2,0)\) and \((6,0)\).

Similarly, it is observed that at \(x = 4\) the curve changes from positive to negative.

Thus, the local maximum is \((4,0)\).

Therefore, the local maximum is \((4,0)\) and the local minimum is \((2,0)\) and \((6,0)\).

(c) Step 5: Determine the intervals for which \(f\) is concave upward and concave downward

It is observed that, the curve is concave upward where \({f^{\prime \prime }}(x) > 0\) and concave downward where \({f^{\prime \prime }}(x) < 0\).

Note that the function \(f\) is concave up whenever the slope of the first derivative is positive and the function \(f\) is concave down whenever the slope of the first derivative is negative.

From the given graph of \({f^\prime }(x)\), it is observed that the slope of the first derivative is positive over the intervals \((1,3),(5,7)\) and \((8,9)\).

Similarly, it is observed that the slope of the first derivative is negative over the intervals \((0,1),(3,5)\) and \((7,8)\).

Thus, \(f\) is concave upward in \((1,3),(5,7)\) and \((8,9)\) and \(f\) is concave downward in \((0,1),(3,5)\) and \((7,8)\).

(d) Step 5: Determine the \(x\)-coordinates of the inflexion point of \(f\)

It is observed that, inflection points occur where the direction of concavity changes.

From part (c), \(f\) is concave upward in \((1,3),(5,7)\) and \((8,9)\) and \(f\) is concave downward in \((0,1),(3,5)\) and \((7,8)\).

Note that, direction of concavity changes at \(x = 1,3,5,7\) and 8.

Thus, the \(x\)-coordinate of the inflection point is \(x = 1,3,5,7\) and 8.