Question

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 -1,11.

x	-1	2	5	8	11

answer attempt 1 out of 2

over successive intervals, the functions

. this evidence is most consistent with the inference that the function is
over the interval -1,11.

Explanation:

Step1: Calculate first differences (slopes between consecutive points)

First, find the differences in \( h(x) \) values between consecutive \( x \) values. The \( x \)-intervals are all length \( 3 \) (since \( 2 - (-1) = 3 \), \( 5 - 2 = 3 \), \( 8 - 5 = 3 \), \( 11 - 8 = 3 \)).

• From \( x = -1 \) to \( x = 2 \): \( \Delta h_1 = 97 - 79 = 18 \)
• From \( x = 2 \) to \( x = 5 \): \( \Delta h_2 = 151 - 97 = 54 \)
• From \( x = 5 \) to \( x = 8 \): \( \Delta h_3 = 241 - 151 = 90 \)
• From \( x = 8 \) to \( x = 11 \): \( \Delta h_4 = 367 - 241 = 126 \)

Step2: Calculate second differences (differences of first differences)

Now, find the differences between these first differences:

• \( \Delta^2 h_1 = 54 - 18 = 36 \)
• \( \Delta^2 h_2 = 90 - 54 = 36 \)
• \( \Delta^2 h_3 = 126 - 90 = 36 \)

Step3: Analyze concavity from second differences

Since the second differences are constant (and positive), this means the first differences (slopes) are increasing. For a function, if the first derivative (slope) is increasing, the function is concave up.

Answer:

Over successive intervals, the function's first differences (slopes) are increasing (with a constant second difference of 36). This evidence is most consistent with the inference that the function is concave up over the interval \([-1, 11]\).