GATE 2018 EC – Question 16
Consider $p(s)=s^3+a_2s^2+a_1s+a_0$ with all real coefficients. It is known that its derivative $p'(s)$ has no real roots. The number of real roots of $p(s)$ is
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Correct answer: (B) 1
Explanation
A real cubic has at least one real root. If $p'$ has no real roots, then $p'$ never changes sign, so $p$ is strictly monotonic and crosses zero exactly once. So it has exactly 1 real root.