Question

Let $f(x)$ and $g(x)$ are two function which are defined and differentiable for all $x\ge x_0$. 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 $\varphi (x)=f(x)-g(x)\Rightarrow {\varphi }^{\prime }(x)=f^{\prime }(x)-g^{\prime }(x)>0$
$\varphi (x)$ is also continuous and derivable in $\left[x_0,x\right]$ using LMVT for $\varphi (x)$ in $\left[x_0,x\right]$
${\varphi }^{\prime }(x)=\frac{\varphi (x)-\varphi \left(x_0\right)}{x-x_0}.$
$\text{ since }{\varphi }^{\prime }(x)=f^{\prime }(x)-g^{\prime }(x)\text{ are }{f}^{\prime }(x)-g^{\prime }(x)>0\text{ for all }x>x_0$
$\therefore {\varphi }^{\prime }(x)>0$
$\text{ hence }\varphi (x)-\varphi \left(x_0\right)>0$
$\varphi (x)>\varphi \left(x_0\right)$
$\left(\varphi \left(x_0\right)=f\left(x_0\right)-g\left(x_0\right)=0\right)$
$f(x)-g(x)>0\Rightarrow$ (D)