If \(f(x) = x^{3} - 4x + p\), and \(f(0)\) and \(f(1)\) are of opposite signs, which of the following…

The range that must hold is $0 < p < 3$. Here $f(0) = p$ and $f(1) = 1 - 4 + p = p - 3$, so opposite signs mean $p(p - 3) < 0$, which happens only when p lies between 0 and 3. The other ranges need not hold: p = 2.5 meets the condition ($f(0) = 2.5$, $f(1) = -0.5$) yet lies outside both $-1 < p < 2$ and $-2 < p < 1$, and no value with $-3 < p < 0$ works at all, because then $f(0)$ and $f(1)$ are both negative.