Select all the intervals where g'(x)<0 and g''(x)>

Questions: Select all the intervals where g'(x)<0 and g''(x)>0. Choose all answers that apply: (A) -4<x<-2 (B) -1<x<1 (C) 3<x<4 (D) None of the above

Transcript text: Select all the intervals where $g^{\prime}(x)<0$ and $g^{\prime \prime}(x)>0$. Choose all answers that apply: (A) $-4

Solution

Solution Steps

Step 1: Analyze $g'(x) < 0$

$g'(x) < 0$ means the function $g(x)$ is decreasing. Looking at the graph, $g(x)$ is decreasing on the intervals $(-4, -2)$ and $(3, 4.5)$.

Step 2: Analyze $g''(x) > 0$

$g''(x) > 0$ means the function $g(x)$ is concave up. Looking at the graph, $g(x)$ is concave up on the intervals $(-2, 1)$ and roughly $(4, 4.5)$.

Step 3: Find the intersection

We need the intervals where both $g'(x) < 0$ and $g''(x) > 0$ are true.
The interval $(-4, -2)$ satisfies $g'(x) < 0$. However, only a portion of $(-2, 1)$ satisfies $g''(x) > 0$. There is no overlap between $(-4, -2)$ and $(-2, 1)$. The interval $(3, 4.5)$ satisfies $g'(x) < 0$, and the interval $(4, 4.5)$ satisfies $g''(x) > 0$. The intersection of these two intervals is $(4, 4.5)$. Since none of the provided answer choices contain this interval, the correct choice is (D)