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 $-1.5 < x < 0$\nc $1 < x < 3$\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 $-1.5 < x < 0$\nc $1 < x < 3$\nd none of the above

Answer

Brief Explanations:

  1. $h'(x) < 0$ means the function is decreasing (moving downward from left to right).
  2. $h''(x) < 0$ means the function is concave down (curving downward, like an upside-down bowl).
  3. Analyze each interval:
    • $-4.5 < x < -3$: The function is decreasing but concave up (curving upward), so $h''(x) > 0$. Does not satisfy both conditions.
    • $-1.5 < x < 0$: The function is increasing (moving upward), so $h'(x) > 0$. Does not satisfy both conditions.
    • $1 < x < 3$: The function is decreasing (moving downward) and concave down (curving downward). This satisfies both $h'(x) < 0$ and $h''(x) < 0$.

Answer:

C. $1 < x < 3$