The range of ab if |a| ≤ 1 and a + b = 1, (a, b ∈ R), is:

Given a + b = 1, substitute b = 1 − a, so

ab = a(1 − a) = −a² + a = −(a − 1/2)² + 1/4.

• Maximum: this is a downward-opening parabola in a with vertex at a = 1/2, giving the largest value ab = 1/4. (a = 1/2 satisfies |a| ≤ 1.)

• Minimum: a is restricted to [−1, 1] by |a| ≤ 1. At the endpoints, a = −1 gives ab = (−1)(1 − (−1)) = (−1)(2) = −2, and a = 1 gives ab = (1)(0) = 0. The smaller is −2, so the minimum is −2.

• As a ranges continuously over [−1, 1], ab takes every value between −2 and 1/4.

Therefore the range of ab is [−2, 1/4].