I have to solve the following exercise but I have no idea how to do it (I'm quite bad in analysis)
On the set $X = \{u \in C^1([0, 1]) : u(0) = 1, u(1) = 0\}$ consider the functional $F : X \to \mathbb{R}$ $$F(u) =\int_0^1(e^{u^{'}(x)}+u^2(x))dx$$ i) Compute the Euler-Lagrange equation associated with the functional F.
ii) Integrate the equation with initial conditions $u(0) = 1$ and $u^{'}(0)=\alpha$ where $\alpha \in \mathbb{R}$ is a parameter.
iii) Prove that F does not have minimum on X
First we compute the EL lagrange equation that, given that $\nabla(u)=u'$ is $(L_\xi)'-L_{u}=0$ and so: $$ L_\xi=e^{u'}\quad L_{u}=2u$$ giving us the equation $$(e^{u'})'-2u=0\iff u''e^{u'}-2u=0 $$.
For the second part we can use the Du Bois-Reymond equation which, for $L(x,u,\nabla(x))=L(u,u')$ is $$u'L_\xi-L=c$$ where c is some constant and allows us to write our problem as $$ u'e^{u'}-e^{u'}-u^2=c$$ and then impose the boundary conditions. This leaves us with a differential equation to solve, and here is the solution https://math.stackexchange.com/a/3613422/764627