Question

Consdier $p\left(s\right)=s^3=a_2^2+a_{1}s+a_0$ with all realcoefficients. It is known that its derivative $p^{\prime }\left(s\right)$ has no real roots. The number of real rotos of $p\left(s\right)$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

By Rolle's theorem we knwo that.
"Between any two real roots of f(x) there exist at least one real root of $f^{\prime }\left(x\right)$"
But in this quesiton it is given that there exist no real root of $P^{\prime }\left(s\right)$ os $P\left(s\right)$ can not have two real roots. Hence only possiblity is thata $P\left(s\right)$ have one real root and two complex roots. [as coefficients of $P\left(s\right)$ are real].