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 maximum
b. a local minimum
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. a local minimum
b. no local extremum
c. a local maximum

Explanation:

Step1: Recall the first - derivative test

If \(f^{\prime}(x)\) changes sign from positive to negative at a critical number \(x = c\), then \(f(x)\) has a local maximum at \(x = c\). If \(f^{\prime}(x)\) changes sign from negative to positive at \(x = c\), then \(f(x)\) has a local minimum at \(x = c\). If \(f^{\prime}(x)\) does not change sign at \(x = c\), then \(f(x)\) has no local extremum at \(x = c\).

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

For \(x\) values slightly less than \(a\), \(f^{\prime}(x)>0\) (the function is increasing). For \(x\) values slightly greater than \(a\), \(f^{\prime}(x)<0\) (the function is decreasing). By the first - derivative test, when the derivative changes from positive to negative at a critical number \(x = a\), the function \(f(x)\) has a local maximum at \(x=a\).

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

The function \(f(x)\) is not differentiable at \(x = b\) (since \(f^{\prime}(x)\) has a non - differentiable point (denoted as \(ND\)) at \(x = b\)). Also, looking at the sign of \(f^{\prime}(x)\) on either side of \(x = b\), the sign of \(f^{\prime}(x)\) does not change (it is negative on both sides of \(x = b\)). So, \(f(x)\) has no local extremum at \(x = b\).

Answer:

For \(x=a\): A. a local maximum
For \(x = b\): B. no local extremum