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
does f(x) have a local minimum, a local maximum, or no local extremum at x = c? 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 a critical number \(x = c\), then \(f(x)\) has a local minimum at \(x = c\). If \(f^{\prime}(x)\) does not change sign at a critical number \(x = c\), then \(f(x)\) has no local extremum at \(x = c\).

Step2: Analyze \(x=a\)

For \(x = a\), \(f^{\prime}(x)>0\) for \(x < a\) (to the left of \(a\)) and \(f^{\prime}(x)<0\) for \(x>a\) (to the right of \(a\)). By the first - derivative test, since the sign of \(f^{\prime}(x)\) changes from positive to negative at \(x = a\), \(f(x)\) has a local maximum at \(x = a\).

Step3: Analyze \(x = b\)

\(x = b\) is a critical number (where \(f^{\prime}(x)\) is non - differentiable). But for \(x\) values close to \(b\) (on both sides), \(f^{\prime}(x)<0\). So the sign of \(f^{\prime}(x)\) does not change at \(x = b\). By the first - derivative test, \(f(x)\) has no local extremum at \(x = b\).

Step4: Analyze \(x = c\)

\(x = c\) is a critical number (where \(f^{\prime}(x)\) is non - differentiable). For \(x\) values close to \(c\) (on both sides), \(f^{\prime}(x)<0\) (to the left of \(c\)) and \(f^{\prime}(x)>0\) (to the right of \(c\)). But wait, no, looking at the sign chart: for \(x\) near \(c\) (left - hand side \(x < c\), \(f^{\prime}(x)<0\) and for \(x>c\), \(f^{\prime}(x)>0\). Since the function \(f(x)\) is continuous, by the first - derivative test (even though \(f^{\prime}(c)\) is non - differentiable), as \(f^{\prime}(x)\) changes sign from negative to positive at \(x = c\), \(f(x)\) has a local minimum at \(x = c\).

Answer:

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