Question

A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1. The second column is labeled f of x with entries 15, negative 5, 0, 5, 0, negative 5.
Which is a valid prediction about the continuous function f(x)?

f(x) ≤ 0 over the interval (–∞, ∞).
f(x) > 0 over the interval (–1, ∞).
f(x) ≥ 0 over the interval [–1, 1].
f(x) < 0 over the interval (0, 2).

Answer 1

f(x) ≥ 0 over the interval [−1, 1].

Reason: f(−1)=0 and f(0)=5, so on [−1,0] (by continuity/IVT) f(x) takes values between 0 and 5, hence ≥0. Similarly f(0)=5 and f(1)=0 imply f(x)≥0 on [0,1]. Thus f(x)≥0 on [−1,1].

(The other choices are false: f(−3)=15 rules out (1); f(1)=0 and f(2)=−5 rule out (2); f(0)=5 and f(1)=0 rule out (4).)