Question

the graph of ( y = f(x) ) is shown. assume ( f(x) ) is only defined over the interval ( -5 leq x leq 3 ).

remember this is the graph of ( y = f(x) ), not the graph of ( y = f(x) )

( f(x) ) will have local minimum(s) at ( x=) (list all ( x )-values corresponding to a local minimum, separated by commas)

( f(x) ) will have local maximum(s) at ( x=) (list all ( x )-values corresponding to a local maximum, separated by commas)

the graph of ( y = f(x) ) have inflection point(s) at ( x=) (list all ( x )-values corresponding to an inflection point, separated by com
question help: video written example

Explanation:

Step1: Recall the first - derivative test for local minima and maxima

• Local minima: A function \(y = f(x)\) has a local minimum at \(x = c\) if \(f^{\prime}(x)\) changes sign from negative to positive at \(x = c\).
• Local maxima: A function \(y = f(x)\) has a local maximum at \(x = c\) if \(f^{\prime}(x)\) changes sign from positive to negative at \(x = c\).
• Inflection points: A function \(y = f(x)\) has an inflection point at \(x = c\) if \(f^{\prime\prime}(x)\) changes sign at \(x = c\), which is equivalent to \(f^{\prime}(x)\) having a local maximum or minimum at \(x = c\).

Step2: Analyze the sign - change of \(y = f^{\prime}(x)\) for local minima

Looking at the graph of \(y = f^{\prime}(x)\), we observe that \(f^{\prime}(x)\) changes sign from negative to positive at \(x=-4\). So, by the first - derivative test, \(f(x)\) has a local minimum at \(x = - 4\).

Step3: Analyze the sign - change of \(y = f^{\prime}(x)\) for local maxima

We see that \(f^{\prime}(x)\) changes sign from positive to negative at \(x = 1\). So, by the first - derivative test, \(f(x)\) has a local maximum at \(x = 1\).

Step4: Analyze the local extrema of \(y = f^{\prime}(x)\) for inflection points

The local extrema of \(y = f^{\prime}(x)\) (where \(f^{\prime\prime}(x)=0\) and \(f^{\prime\prime}(x)\) changes sign) occur at \(x=-2\) (local minimum of \(f^{\prime}(x)\)) and \(x = 4\) (local maximum of \(f^{\prime}(x)\)). So, the inflection points of \(y = f(x)\) are at \(x=-2,4\).

Answer:

• \(f(x)\) will have local minimum(s) at \(x=-4\)
• \(f(x)\) will have local maximum(s) at \(x = 1\)
• The graph of \(y = f(x)\) have inflection point(s) at \(x=-2,4\)