IIT JEE 1983 Maths - MCQ Question 7

Let

$\begin{aligned} f^{'}(x) & = 3ax^2+2bx+c \\ \displaystyle\int f^{'}(x) dx & = ax^3+bx^2+cx+C \\ f(x) & = ax^3+bx^2+cx+C \\ f(0) & = C\hspace{2.5mm},\hspace{2.5mm} f(1) =C \end{aligned}$

From Rolle's theorem $f^{'}(x)$ must be zero between $0$ and $1$ . Since $f(0)=f(1) = C$ and $f(x)$ is continuous .

Hence , there is atleast one root of $f^{'}(x)$ in $\left[ 0,1 \right]$