KVPY Sample Paper KVPY Stream-SX Model Paper-13

If \[g\,(x)\]is a differential real valued function satisfying \[g''(x)-3g'(x)>3,x\ge 0\] and \[g'(0)=-1,\] then \[g\,(x)+x\]for \[x>0\]is

A) an increasing function

B) a decreasing function

C) a constant function

D) None of these

Correct Answer: A

Solution :

We have, \[g''(x)-3g'(x)>3\]

\[\Rightarrow \]\[\frac{g''(x)}{g'(x)+1}>3\]

On integrating, we get \[\log \,(g'(x)+1)>3x\]

\[\Rightarrow \]\[g'(x)+1>{{e}^{3x}}\]

\[g'(x)+1>0\forall x>0\]

\[\therefore \]\[g(x)+x\]is an increasing function.