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 = b? choose the correct answer below.

does f(x) have a local minimum, a local maximum, or no local extremum at x = c? choose the correct answer below.

does f(x) have a local minimum, a local maximum, or no local extremum at x = d? choose the correct answer below.

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 \(x = b\)

At \(x = b\), \(f^{\prime}(x)\) does not change sign (the sign of \(f^{\prime}(x)\) is negative on both sides of \(x = b\) (considering the non - differentiable point)). So, by the first - derivative test, \(f(x)\) has no local extremum at \(x = b\).

Step3: Analyze \(x = c\)

At \(x = c\), \(f^{\prime}(x)\) changes sign from negative to positive. So, by the first - derivative test, \(f(x)\) has a local minimum at \(x = c\).

Step4: Analyze \(x = d\)

At \(x = d\), \(f^{\prime}(x)\) changes sign from positive to positive (the sign of \(f^{\prime}(x)\) is positive on both sides of \(x = d\)). So, by the first - derivative test, \(f(x)\) has no local extremum at \(x = d\).

Answer:

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