Question

Let $f ( x )$ and $g ( x )$ are two function which are defined and differentiable for all $x \geq x _0$. If $f \left( x _0\right)= g \left( x _0\right)$ and $f ^{\prime}( x )> g ^{\prime}( x )$ for all $x > x _0$ then

• A. $f(x)< g(x)$ for some $x>x_0$
• B. $f(x)=g(x)$ for some $x>x_0$
• C. $f(x) >g(x)$ only for some $x>x_0$
• D. $f(x) >g(x)$ for all $x>x_0$

Answer 1

Answer: (D)

Consider $ \phi(x)=f(x)-g(x) \Rightarrow \phi^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x)>0$
$\phi( x )$ is also continuous and derivable in $\left[ x _0, x \right]$ using LMVT for $\phi( x )$ in $\left[ x _0, x \right]$
$\phi^{\prime}( x )=\frac{\phi( x )-\phi\left( x _0\right)}{ x - x _0} .$
$\text { since } \phi^{\prime}(x)=f^{\prime}(x)-g^{\prime}(x) \text { are } f^{\prime}(x)-g^{\prime}(x)>0 \text { for all } x >x_0$
$\therefore \phi^{\prime}( x )>0 $
$\text { hence } \phi(x)-\phi\left(x_0\right)>0 $
$\phi( x )>\phi\left( x _0\right) $
$\left(\phi\left( x _0\right)= f \left( x _0\right)- g \left( x _0\right)=0\right)$
$f(x)-g(x)>0 \Rightarrow $ (D)