Question

f(x) is continuous on (-∞,∞) and has critical numbers at x = a, b, c, and d. use the sign chart for f(x) to determine whether f has a local maximum, a local minimum, or neither at each critical number.
does f(x) have a local minimum, a local maximum, or no local extremum at x = a? choose the correct answer below.
a. a local minimum
b. a local maximum
c. no local extremum
does f(x) have a local minimum, a local maximum, or no local extremum at x = b? choose the correct answer below.
a. no local extremum
b. a local minimum
c. a local maximum
does f(x) have a local minimum, a local maximum, or no local extremum at x = c? choose the correct answer below.
a. no local extremum
b. a local minimum
c. a local maximum

Explanation:

Step1: Analyze the sign of \(f^{\prime}(x)\) around \(x = a\)

To the left of \(x=a\), \(f^{\prime}(x)<0\) (function is decreasing). To the right of \(x = a\), \(f^{\prime}(x)>0\) (function is increasing). By the first - derivative test, if \(f^{\prime}(x)\) changes sign from negative to positive at a critical number \(x = c\), then \(f(x)\) has a local minimum at \(x = c\).

Step2: Analyze the sign of \(f^{\prime}(x)\) around \(x = b\)

A critical number \(x = b\) is a non - differentiable point. But for a local extremum at a non - differentiable point \(x = c\), the function must change its increasing/decreasing behavior. Here, the function is increasing on both sides of \(x = b\) (since \(f^{\prime}(x)>0\) on both sides of \(x = b\) in the relevant intervals). So, there is no local extremum at \(x = b\).

Step3: Analyze the sign of \(f^{\prime}(x)\) around \(x = c\)

A critical number \(x = c\) is a non - differentiable point. The function is increasing on both sides of \(x = c\) (since \(f^{\prime}(x)>0\) on both sides of \(x = c\) in the relevant intervals). So, there is no local extremum at \(x = c\).

Answer:

For \(x = a\): A. a local minimum
For \(x = b\): A. no local extremum
For \(x = c\): A. no local extremum