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

Explanation:

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

Before \(x=a\), \(f^{\prime}(x)<0\) (function is decreasing). After \(x = a\), \(f^{\prime}(x)>0\) (function is increasing).

Step2: Apply the first - derivative test

By the first - derivative test, if \(f^{\prime}(x)\) changes sign from negative to positive at a critical number \(x = c\), then \(f(x)\) has a local minimum at \(x = c\). Since \(f^{\prime}(x)\) changes from negative to positive at \(x=a\), \(f(x)\) has a local minimum at \(x = a\).

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

\(x = b\) is not a critical number (because \(f^{\prime}(x)\) does not change its non - zero sign in a neighborhood of \(x = b\), and \(f^{\prime}(b)\) is not defined in the sense of being a critical number for the derivative - based extremum test. A critical number for the first - derivative test is a point where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined, but for the first - derivative test to detect an extremum, if \(f^{\prime}(x)\) is undefined at a point \(x = c\), the sign of \(f^{\prime}(x)\) on either side of \(x = c\) must change. Here, the sign of \(f^{\prime}(x)\) does not change around \(x = b\) (it is positive on both sides near \(x = b\)).

Answer:

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