Question

If $f\left(x\right)$ and $g\left(x\right)$ are differentiable functions for $0\le x\le 1$ such that $f\left(0\right)=10$, $g\left(0\right)=2$, $f\left(1\right)=2$, $g\left(1\right)=4$, then in the interval $(0,1)$,

• A. $f^{\prime }\left(x\right)=0$ for at least one $x$
• B. $f^{\prime }\left(x\right)+4g^{\prime }\left(x\right)=0$ for at least one $x$.
• C. $f^{\prime }\left(x\right)=2g^{\prime }\left(x\right)$ for at most one $x$
• D. none of these

Answer 1

The correct option is B $f^{\prime }\left(x\right)+4g^{\prime }\left(x\right)=0$ for at least one $x$.

By Cauchy's mean value theorem,
$\frac{f^{\prime }\left(c\right)}{g^{\prime }\left(c\right)}=\frac{f\left(1\right)-f\left(0\right)}{g\left(1\right)-g\left(0\right)}$
$\frac{f^{\prime }\left(c\right)}{g^{\prime }\left(c\right)}=-4$
$f^{\prime }\left(c\right)=-4g^{\prime }\left(c\right)$
$f^{\prime }\left(c\right)+4g^{\prime }\left(c\right)=0$
$f^{\prime }\left(x\right)+4g^{\prime }\left(x\right)=0$ for at least one $x$.