Question

determining a local maximum and minimum. analyze the table of values for the continuous function, f(x), to complete the statements. a local maximum occurs over the interval. a local minimum occurs over the interval

Explanation:

Step1: Understand local maximum and minimum

A local maximum is a point where the function value is greater than the values at neighboring points, and a local minimum is a point where the function value is less than the values at neighboring points.

Step2: Analyze the table

We see that when \(x=-1\), \(f(-1) = 2\). The values of \(f(x)\) for \(x=-2\) and \(x = 0\) are \(-1\), which are less than \(f(-1)\). So a local maximum occurs around \(x=-1\), and the interval could be \((-2,0)\).

Step3: Find local minimum

We note that when \(x = 1\), \(f(1)=-4\). The values of \(f(x)\) for \(x = 0\) and \(x=2\) are \(-1\), which are greater than \(f(1)\). So a local minimum occurs around \(x = 1\), and the interval could be \((0,2)\).

Answer:

A local maximum occurs over the interval \((-2,0)\)
A local minimum occurs over the interval \((0,2)\)