Now as u′(0)>0, it is clear that the function will be convex throughout the interval [0,∞), thus u(x),u′(x),u"(x) will all be positive in the interval. Both statements follow.
@DevanshBhardwaj How did you claimed the convexity?
It would be possible only if u''(x)>0 for all x in R.
As $u"(0)>0$ and $u'(0)>0$, thus, $\int_0^xu'(t)\,dt>0$ for all $t>0$. I hope this much seems clear.
@AasthaChoudhary Convexity in the interval is what we are concerned about, not throughout $R$. Suppose the function $x^3$. It is convex throughout $R^+$ even though it is concave in $R^-$. So not necessary throughout R.
Well I think I should make it a little formal, which I am not able to think how right now. Indeed I understand why you are not able to understand me. Bye
