We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $a \in \mathbb { R } ^ { 2 }$ and $G ^ { a }$ be the map from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ defined by $$\forall p \in \mathbb { R } ^ { 2 } , \quad G ^ { a } ( p ) = \| F ( p ) - a \| ^ { 2 }$$
[III.C.1)] If $p$ and $h$ are in $\mathbb { R } ^ { 2 }$, compute $d G ^ { a } { } _ { p } ( h )$.
[III.C.2)] Show that $G ^ { a } ( p ) \rightarrow + \infty$ when $\| p \| \rightarrow + \infty$.
[III.C.3)] Deduce that $G ^ { a }$ attains a global minimum on $\mathbb { R } ^ { 2 }$ at a point $p _ { 0 }$.
[III.C.4)] Show that $F \left( p _ { 0 } \right) = a$.
We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$
$$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$
Let $a \in \mathbb { R } ^ { 2 }$ and $G ^ { a }$ be the map from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ defined by
$$\forall p \in \mathbb { R } ^ { 2 } , \quad G ^ { a } ( p ) = \| F ( p ) - a \| ^ { 2 }$$
\begin{enumerate}
\item[III.C.1)] If $p$ and $h$ are in $\mathbb { R } ^ { 2 }$, compute $d G ^ { a } { } _ { p } ( h )$.
\item[III.C.2)] Show that $G ^ { a } ( p ) \rightarrow + \infty$ when $\| p \| \rightarrow + \infty$.
\item[III.C.3)] Deduce that $G ^ { a }$ attains a global minimum on $\mathbb { R } ^ { 2 }$ at a point $p _ { 0 }$.
\item[III.C.4)] Show that $F \left( p _ { 0 } \right) = a$.
\end{enumerate}