Mathematics - Application of Derivative Question with Solution | TestHub

Let

$h\left(x\right)=f\left(x\right)-2g\left(x\right)$

$h\left(0\right)=f\left(0\right)-2g\left(0\right)$

$h\left(0\right)=2-2\left(0\right)$   $\left[\because f\left(0\right)=2,g\left(0\right)=0\right]$

$h\left(0\right)=2$

$h\left(1\right)=f\left(1\right)-2g\left(1\right)$

$h\left(1\right)=6-2\left(2\right)$    $\left[\because f\left(1\right)=6,g\left(1\right)=2\right]$

$h\left(1\right)=2$

$h\left(0\right)=h\left(1\right)$

$h^{\text{'}}\left(x\right)=f^{\text{'}}\left(x\right)-2g^{\text{'}}\left(x\right)$

By Rolle's theorem $h^{\text{'}}\left(c\right)=0$

$\therefore f^{\text{'}}\left(c\right)-2g^{\text{'}}\left(c\right)=0,f^{\text{'}}\left(c\right)=2g^{\text{'}}\left(c\right)$