Question

predicting key aspects using a table
x -4 -3 -2 -1 0 1 2 3 4
f(x) -54 -20 -4 0 -2 -4 0 16 50
which interval contains a local maximum for this function?
which interval contains a local minimum for this function?

Explanation:

Step1: Recall local - max/min definition

A local maximum occurs when the function value is greater than the values at nearby points, and a local minimum occurs when the function value is less than the values at nearby points.

Step2: Analyze the function values

Looking at the table of \(f(x)\) values:

• As \(x\) goes from \(-2\) to \(0\), \(f(x)\) changes from \(-4\) to \(-2\) to \(-4\). The function value at \(x = 0\) is \(-2\) which is greater than the values of \(f(x)\) at \(x=- 1\) and \(x = 1\). So, the interval \((-1,1)\) contains a local maximum.
• As \(x\) goes from \(-1\) to \(1\), \(f(x)\) changes from \(0\) to \(-2\) to \(-4\). The function value at \(x=-1\) is \(0\) and at \(x = 1\) is \(-4\), and the value at \(x = 0\) is \(-2\). Also, as \(x\) goes from \(1\) to \(3\), \(f(x)\) changes from \(-4\) to \(0\) to \(16\). The function value at \(x = 1\) is \(-4\) which is less than the values of \(f(x)\) at \(x=0\) and \(x = 2\). So, the interval \((0,2)\) contains a local minimum.

Answer:

• Which interval contains a local maximum for this function? \((-1,1)\)
• Which interval contains a local minimum for this function? \((0,2)\)