Question

Statement $1:$ If $g\left(x\right)$ is a differentiable function, $g\left(2\right)\ne 0,$ $g(-2)\ne 0$ and Rolle's theorem is not applicable to $f\left(x\right)$ $=\frac{x^2-4}{g\left(x\right)}$ in $[-2,2]$, then $g\left(x\right)$ has at least one root in $(-2,2)$.
Statement $2:$ If $f\left(a\right)=f\left(b\right)$, then Rolle's theorem is applicable for $x\in (a,b)$

• A. Only Statement $1$ is true
• B. Only Statement $2$ is true
• C. Both statements are true
• D. Both statements are false

Answer 1

The correct option is A Only Statement $1$ is true

Statement $1:$
Given : $g\left(x\right)$ is a differentiable function and Rolle's theorem is not applicable to $f\left(x\right)$ in $(-2,2),$ but $f(-2)=f\left(2\right)=0$.
It implies that either $f\left(x\right)$ is discontinuous or not differentiable at atleast one point in $(-2,2).$
So, $g\left(x\right)=0$ has atleast one root in $(-2,2)$.
Hence, statement $1$ is correct.
Statement $2:$
Since $f\left(x\right)$ must be continuous in $[a,b]$ and differentiable in $(a,b)$ is not given.
$\therefore$ Statement $2$ is wrong.