select all the intervals where $h(x) > 0$ and $h(x) < 0$.\nchoose all answers that apply:\na $-4.5 < x <…

select all the intervals where $h(x) > 0$ and $h(x) < 0$.\nchoose all answers that apply:\na $-4.5 < x < -3$\nb $-3 < x < -1$\nc $-1 < x < 1$\nd none of the above

select all the intervals where $h(x) > 0$ and $h(x) < 0$.\nchoose all answers that apply:\na $-4.5 < x < -3$\nb $-3 < x < -1$\nc $-1 < x < 1$\nd none of the above

Answer

Explanation:

Step1: Interpret $h'(x) > 0$

$h'(x) > 0$ means the graph of $h(x)$ is increasing (rising as $x$ increases).

Step2: Interpret $h''(x) < 0$

$h''(x) < 0$ means the graph of $h(x)$ is concave down (curving downward, like an upside-down bowl).

Step3: Analyze each interval

  • Interval $-4.5 < x < -3$: The graph is decreasing and concave down. $h'(x) < 0$, so this fails.
  • Interval $-3 < x < -1$: The graph is decreasing and concave up. $h'(x) < 0$, so this fails.
  • Interval $-1 < x < 1$: The graph is increasing and concave up. $h''(x) > 0$, so this fails.

Step4: Identify valid interval

The only interval where $h(x)$ is increasing and concave down is $(3, 5)$ (not listed in options).

Answer:

D. None of the above