Question

If \(f(x) = x^3 - 4x + p\), and \(f(0)\) and \(f(1)\) are of opposite signs, then which of the following is necessarily true?

• A. \(-1<p<2\)
• B. \(0<p<3\)
• C. \(-2<p<1\)
• D. \(-3<p<0\)

Hint:

For sign-change problems, evaluate the function at key points and apply the intermediate value theorem.

Answer 1

The Correct Option is A

We know \(f(0) = p\) and \(f(1) = 1 - 4 + p = -3 + p\). For \(f(0)\) and \(f(1)\) to have opposite signs, we must have: \[ p>0 \quad \text{and} \quad -3 + p<0 \] Thus, the solution is \( -1<p<2\). \[ \boxed{-1<p<2} \]