Question

A 2-column table with 7 rows. The first column is labeled x with entries negative 5, negative 3, negative 1, 1, 3, 5, 7. The second column is labeled f of x with entries 8, 4, 0, negative 2, negative 2, 0, 4.
Which is a valid prediction about the continuous function f(x)?

f(x) ≥ 0 over the interval [5, ∞).
f(x) ≤ 0 over the interval [–1, ∞).
f(x) > 0 over the interval (–∞, 1).
f(x) < 0 over the interval (–∞. –1).

Answer 1

None of them.

Brief reasons:
- A (f(x) ≥ 0 on [5, ∞)): continuity and the table only force nonnegativity on [5,7] (f(5)=0, f(7)=4). For x>7 the function could later go negative, so [5,∞) is not guaranteed.
- B (f(x) ≤ 0 on [−1, ∞)): false because f(7)=4>0.
- C (f(x) > 0 on (−∞,1)): false because f(−1)=0 (not >0).
- D (f(x) < 0 on (−∞,−1)): false because f(−3)=4>0.

So no given statement must be true for every continuous f fitting the table.