Question

the graph of the derivative, $f(x)$, of a function $f(x)$ is shown below.

on what intervals is $f(x)$ increasing?

at what values of $x$ does $f(x)$ have a local maximum?

on what intervals is $f(x)$ concave up?

Explanation:

Step1: Determine where \(f(x)\) is increasing

A function \(f(x)\) is increasing when \(f^{\prime}(x)>0\). Looking at the graph of \(y = f^{\prime}(x)\), \(f^{\prime}(x)>0\) on the intervals \((0,4)\) and \((6,8)\).

Step2: Find local maximum of \(f(x)\)

A local maximum of \(f(x)\) occurs where \(f^{\prime}(x)\) changes from positive to negative. At \(x = 4\) and \(x=8\), \(f^{\prime}(x)\) changes from positive to negative.

Step3: Determine where \(f(x)\) is concave up

A function \(f(x)\) is concave up when \(f^{\prime\prime}(x)>0\). Since \(f^{\prime\prime}(x)\) is the derivative of \(f^{\prime}(x)\), \(f(x)\) is concave up where \(f^{\prime}(x)\) is increasing. Looking at the graph of \(y = f^{\prime}(x)\), \(f^{\prime}(x)\) is increasing on the intervals \((1,2)\) and \((5,7)\).

Answer:

• \(f(x)\) is increasing on \((0,4)\cup(6,8)\)
• \(f(x)\) has local maxima at \(x = 4\) and \(x = 8\)
• \(f(x)\) is concave up on \((1,2)\cup(5,7)\)