Question

Let f(x) and g(x) are defined and differentiable for $x\ge x_0$ and $f\left(x_0\right)=g\left(x_0\right),f^{\prime }\left(x\right)>g^{\prime }\left(x\right)$ for $x>x_0$ then

• A. $f\left(x\right)<g\left(x\right)$ for some $x>x_0$
• B. $f\left(x\right)=g\left(x\right)$ for some $x>x_0$
• C. $f\left(x\right)>g\left(x\right)$ for some $x>x_0$
• D. none of these

Answer 1

Answer: (C)

$f\left(x\right)>g\left(x\right)$ for some $x>x_0$

Let
$h\left(x\right)=f\left(x\right)-g\left(x\right)$
Hence
$h^{\prime }\left(x\right)$
$=f^{\prime }\left(x\right)-g^{\prime }\left(x\right)$
Now for $x>x_0$
$f^{\prime }\left(x\right)>g^{\prime }\left(x\right)$
Or
$f^{\prime }\left(x\right)-g^{\prime }\left(x\right)>0$
Or
$h^{\prime }\left(x\right)>0$ for $x>x_0$
Hence
$h\left(x\right)$ is an increasing function for all $x>x_0$
Or
$h\left(x\right)>0$ for $x>x_0$
Or
$f\left(x\right)-g\left(x\right)>0$ for $x>x_0$
Or
$f\left(x\right)>g\left(x\right)$ for $x>x_0$.