Question: The table below shows values for some polynomial function $h$ at selected values of $x$. Fill in the guided sentence below to make a valid claim regarding the concavity of the function over the interval $[-12,-8]$.
x -12 -11 -10 -9 -8
h(x) -171 -97 -59 -45 -43
Answer
Attempt 1 out of 3
Over successive intervals, the function’s
. This evidence is most consistent with the inference that the function is
over the interval $[-12,-8]$.
1. **State the problem:** We are given values of a polynomial function $h(x)$ at $x = -12, -11, -10, -9, -8$ and asked to analyze the concavity of $h$ over the interval $[-12,-8]$.
2. **Recall the concept of concavity:**
- A function is **concave up** on an interval if its second derivative $h''(x) > 0$ there.
- A function is **concave down** on an interval if $h''(x) < 0$ there.
3. **Use the given data to estimate concavity:**
- First, calculate the first differences (approximate first derivative) between consecutive points:
$$
h'(-11) \approx \frac{h(-11) - h(-12)}{-11 - (-12)} = \frac{-97 - (-171)}{1} = 74
$$
$$
h'(-10) \approx \frac{-59 - (-97)}{1} = 38
$$
$$
h'(-9) \approx \frac{-45 - (-59)}{1} = 14
$$
$$
h'(-8) \approx \frac{-43 - (-45)}{1} = 2
$$
4. **Calculate the second differences (approximate second derivative):**
- Differences of the first differences:
$$
h''(-10) \approx \frac{38 - 74}{1} = -36
$$
$$
h''(-9) \approx \frac{14 - 38}{1} = -24
$$
$$
h''(-8) \approx \frac{2 - 14}{1} = -12
$$
5. **Interpretation:**
- The second differences are negative and increasing (less negative) but still negative.
- This means the function's slope is decreasing, indicating the function is **concave down** over the interval.
6. **Fill in the sentence:**
Over successive intervals, the function’s **rate of change of slope decreases (second differences are negative)**. This evidence is most consistent with the inference that the function is **concave down** over the interval $[-12,-8]$.