Question

A 2-column table with 6 rows. The first column is labeled t with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries negative 12, m, 4, 0, negative 4, negative 2.
If the table of the function contains exactly two potential turning points, one with an input value of –1, which statement best describes all possible values of m?

m ≥ –12
–12 < m < 4
m ≤ 4
m ≥ 4 or m ≤ –12

Answer 1

Compute turning points at interior inputs t = -2, -1, 0, 1.

- f(-3) = -12, f(-2) = m, f(-1) = 4, f(0) = 0, f(1) = -4, f(2) = -2.
- t = -1 is a turning point exactly when 4 is greater than both neighbors, i.e. when m < 4.
- t = 1 is always a turning point (−4 < 0 and −4 < −2).
- t = -2 is a turning point iff m > 4 or m < −12.
- t = 0 is never a turning point.

To have exactly two turning points with one at t = −1, we need m < 4 and we must avoid making t = −2 a turning point, so m must satisfy −12 ≤ m < 4.

Among the given choices the best match is: −12 < m < 4 (choice 2). (Note: m = −12 also gives exactly two turning points, so the exact set is −12 ≤ m < 4.)