Based on the extreme value theorem, what is the maximum value of f of x is equal to negative x squared plus 6 x over the interval left bracket 1 comma 4 right bracket ?

The function f(x) = −x² + 6x is a downward-opening parabola, so its maximum occurs at its vertex. The vertex has x-coordinate −6 ÷ (2 × −1) = 3, which lies in the interval [1, 4]. Evaluating gives f(3) = −9 + 18 = 9, so the maximum value is 9, choice B.

AI-written and checked against the verified answer.