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

Explanation:

Step1: Recall the first - derivative test

If \(f^{\prime}(x)\) changes 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 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 \(a\), \(f^{\prime}(x)>0\) (since \(f^{\prime}(x)\) is \(+++\) before \(x = a\)). For \(x\) values slightly greater than \(a\), \(f^{\prime}(x)<0\) (since \(f^{\prime}(x)\) is \(-- -\) after \(x = a\)).

Answer:

A. a local maximum