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

Explanation:

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

To the left of \(x=a\), \(f^{\prime}(x)<0\) (negative). To the right of \(x = a\), \(f^{\prime}(x)>0\) (positive).

Step2: Apply the First - Derivative Test

The First - Derivative Test states that if \(f^{\prime}(x)\) changes sign from negative to positive at a critical number \(x = c\), then \(f(x)\) has a local minimum at \(x = c\).

Answer:

A. a local minimum