Question

8 fill in the blank 6 points the figure shows the graph of the first derivative ( f^{prime}(x) ) of a function ( f ) defined on the interval ( (0,8) ). the ( x )-value(s) where ( f ) has a local minimum: type your answer. interval(s) where the graph of ( f ) is concave down: type your answer. the ( x )-coordinate(s) of the inflection point(s) of ( f ): type your answer. (if multiple, separate with commas)

Explanation:

Local Minimum:

• Step1: Recall the first - derivative test

A function \(y = f(x)\) has a local minimum at a point \(x = c\) if \(f^{\prime}(x)\) changes sign from negative to positive at \(x = c\).
Looking at the graph of \(y=f^{\prime}(x)\), we see that \(f^{\prime}(x)\) changes sign from negative to positive at \(x = 4\).

Concave Down:

• Step1: Recall the concavity rule

The graph of \(y = f(x)\) is concave down when \(f^{\prime\prime}(x)<0\). Since \(f^{\prime\prime}(x)\) is the derivative of \(f^{\prime}(x)\), we need to find where \(y = f^{\prime}(x)\) is decreasing.
The function \(y = f^{\prime}(x)\) is decreasing on the intervals \((1,3)\) and \((6,7)\)

Inflection Points:

• Step1: Recall the inflection - point rule

Inflection points of \(y = f(x)\) occur where \(f^{\prime\prime}(x)\) changes sign, which is equivalent to where \(y = f^{\prime}(x)\) has a local maximum or minimum.
The function \(y = f^{\prime}(x)\) has a local maximum at \(x = 1\) and \(x = 6\) and a local minimum at \(x = 4\)

Answer:

• The \(x\) - value(s) where \(f\) has a local minimum: \(4\)
• Interval(s) where the graph of \(f\) is concave down: \((1,3),(6,7)\)
• The \(x\) - coordinate(s) of the inflection point(s) of \(f\): \(1,4,6\)