Question

q8 in the function shown in the chart below, at which values of x are there relative minima or maxima?
xf(x)
-3 - 2
-2 2
-1 0
0 - 2
1 2
2 18
3 52
2 and - 2
-3 and 3
-2 and 0
-3, 0, and 3

Explanation:

Step1: Understand relative minima and maxima

A relative maximum is a point where the function changes from increasing to decreasing. A relative minimum is a point where the function changes from decreasing to increasing.

Step2: Analyze the function values

• For \(x = - 2\): The function value \(f(-3)=-2\), \(f(-2) = 2\), \(f(-1)=0\). The function increases from \(x=-3\) to \(x = - 2\) (\(f(-3)<f(-2)\)) and then decreases from \(x=-2\) to \(x=-1\) (\(f(-2)>f(-1)\)), so \(x=-2\) is a relative maximum.
• For \(x = 0\): The function value \(f(-1) = 0\), \(f(0)=-2\), \(f(1)=2\). The function decreases from \(x=-1\) to \(x = 0\) (\(f(-1)>f(0)\)) and then increases from \(x = 0\) to \(x=1\) (\(f(0)<f(1)\)), so \(x = 0\) is a relative minimum.

Answer:

-2 and 0