Consider the following statement about the polynomial p(x), where a and b are real numbers with a < b:
(×) There exists a number c with a < c < b such that p'(c) = 0.
Which one of the following is true?
- AThe condition p(a) = p(b) is necessary and sufficient for (×)
- BThe condition p(a) = p(b) is necessary but not sufficient for (×)
- CThe condition p(a) = p(b) is sufficient but not necessary for (×)
- DThe condition p(a) = p(b) is not necessary and not sufficient for (×)
Show the answer and worked solution
answer · C
- AThe condition p(a) = p(b) is necessary and sufficient for (×)
- BThe condition p(a) = p(b) is necessary but not sufficient for (×)
- CThe condition p(a) = p(b) is sufficient but not necessary for (×)
- DThe condition p(a) = p(b) is not necessary and not sufficient for (×)
Sufficiency is Rolle's theorem: a polynomial is continuous and differentiable, so if it takes the same value at a and b it must turn somewhere strictly between them, giving a c with p'(c) = 0. Necessity fails, and one example is enough: take p(x) = x3 on a = −1, b = 2. Here p'(0) = 0 with −1 < 0 < 2, so (×) holds, yet p(−1) = −1 ≠ 8 = p(2).