Question

If $f\left(x\right)$ and $g\left(x\right)$ are differentiable functions for $0\le x\le 1$ such that $f\left(0\right)=2,g\left(0\right)=0,f\left(1\right)=6,g\left(1\right)=2$, then in the interval $(0,1)$

• A. $f^{\prime }\left(x\right)=0$ for all $x$
• B. $f^{\prime }\left(x\right)=2g^{\prime }\left(x\right)$ for at least one $x$
• C. $f^{\prime }\left(x\right)=2g^{\prime }\left(x\right)$ for most one $x$
• D. None of these

Answer 1

The correct option is B $f^{\prime }\left(x\right)=2g^{\prime }\left(x\right)$ for at least one $x$

Applying Lagrange's theorem, we get

$\frac{f\left(1\right)-f\left(0\right)}{1-0}=f^{\prime }\left(c\right)$

$f^{\prime }\left(c\right)=4$ ...(i)

Also

$\frac{g\left(1\right)-g\left(0\right)}{1-0}=g^{\prime }\left(c\right)$

$g^{\prime }\left(c\right)=2$...(ii)

Where $0\le c\le 1$.

Hence from i and ii

$f^{\prime }\left(c\right)=2g^{\prime }\left(c\right)$ where $0\le c\le 1$.