Question

x | f(x)
-6 | 8
-4 | 2
-2 | 0
0 | -2
2 | -1
4 | 0
6 | 4
which is a possible turning point for the continuous function f(x)?
(-2, 0)
(0, -2)
(2, -1)
(4, 0)

Explanation:

Step1: Analyze function's trend

First, observe the values of \( f(x) \) as \( x \) increases. From \( x = -6 \) to \( x = 2 \), \( f(x) \) decreases (from 8 to -1), and from \( x = 2 \) to \( x = 6 \), \( f(x) \) increases (from -1 to 4). A turning point is where the function changes from decreasing to increasing or vice - versa.

Step2: Check the options

• For option \((-2,0)\): Around \( x=-2 \), the function is still decreasing (from \( x = - 4,f(x)=2\) to \( x=-2,f(x) = 0\) and then to \( x = 0,f(x)=-2\)), so it's not a turning point.
• For option \((0,-2)\): Around \( x = 0\), the function is still decreasing (from \( x=-2,f(x)=0\) to \( x = 0,f(x)=-2\) and then to \( x = 2,f(x)=-1\)), so it's not a turning point.
• For option \((2,-1)\): Before \( x = 2\) (from \( x=-6\) to \( x = 2\)), the function is decreasing, and after \( x = 2\) (from \( x = 2\) to \( x = 6\)), the function is increasing. So at \( x = 2\), the function changes from decreasing to increasing, so \((2,-1)\) is a possible turning point.
• For option \((4,0)\): Around \( x = 4\), the function is increasing (from \( x = 2,f(x)=-1\) to \( x = 4,f(x)=0\) and then to \( x = 6,f(x)=4\)), so it's not a turning point.

Answer:

(2, -1)