We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ For $x \in \mathbb { R } _ { + } ^ { * }$, we set $\Phi ( x ) = \frac { 4 \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x } - \int _ { 0 } ^ { x } \frac { \mathrm { e } ^ { t / 2 } } { \sqrt { t } } \mathrm {~d} t$. Show that $\Phi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$, that $\lim _ { x \rightarrow 0 } \Phi ( x ) = 0$ and that, for all $x > 0 , \Phi ^ { \prime } ( x ) \geqslant 0$. Deduce that $\Phi ( x ) \geqslant 0$ for all $x > 0$.
We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying
$$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$
For $x \in \mathbb { R } _ { + } ^ { * }$, we set $\Phi ( x ) = \frac { 4 \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x } - \int _ { 0 } ^ { x } \frac { \mathrm { e } ^ { t / 2 } } { \sqrt { t } } \mathrm {~d} t$. Show that $\Phi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$, that $\lim _ { x \rightarrow 0 } \Phi ( x ) = 0$ and that, for all $x > 0 , \Phi ^ { \prime } ( x ) \geqslant 0$. Deduce that $\Phi ( x ) \geqslant 0$ for all $x > 0$.