Question

Let $f\left(x\right)$ be a non-constant polynomial such that $f\left(a\right)=f\left(b\right)=f\left(c\right)=2.$ Then the minimum number of roots of the equation $f^{"}\left(x\right)=0$ in $x\in \left(a , c\right)$ is/are

• A. $2$
• B. $1$
• C. $0$
• D. $3$

Answer 1

Answer: (B)

As $f\left(a\right)=f\left(b\right)=f\left(c\right),$ then by Rolle’s theorem, we get,
$f\left(c_{1}\right)=f\left(c_{2}\right)=0$ $\Rightarrow $ for some $c_{1}\in \left(a , b\right)$ & $c_{2}\in \left(b , c\right)$
Again, applying Rolle’s theorem to function $y=f^{'}\left(x\right),$
we have $f^{"}\left(c\right)=0$ for some $c_{3}\in \left(c_{1} , c_{2}\right)$
Hence, the equation $f^{"}\left(x\right)=0$ has at least one root.