Question

Let $f\left(x\right)$ and $g\left(x\right)$ are two functions which are defined and differentiable for all $x\ge x_0$. If $f\left(x_0\right)=g\left(x_0\right)$ and $f^{\prime }\left(x\right)>g^{\prime }\left(x\right)$ for all $x>x_0$ 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. $f\left(x\right)>g\left(x\right)$ only for some $x>x_0$
• D. $f\left(x\right)>g\left(x\right)$ for all $x>x_0$

Answer 1

The correct option is D $f\left(x\right)>g\left(x\right)$ for all $x>x_0$

As $f^{{}^{\prime }}\left(x\right)>g^{{}^{\prime }}\left(x\right)$ for$x>x_0$
That implies slope of $f\left(x\right)$ is greater than slope of $g\left(x\right)$
Then for $x>x_0\Rightarrow f\left(x\right)>g\left(x\right)$
Hence, option 'D' is correct.