Infimum and supremum of $\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$

Question

Find infimum and supremum of $$\phi[x]=\int_{0}^{1} e^{x(t)}(x'(t))^{2} \; dt$$ where $x \in C^{1}[0,1]$ and $x(0)=0$ and $x(1)=\log 4$.

It's easy to show that $\sup \phi[x]=\infty$, but what about $\inf$? I suppose that $\inf \phi[x]=0$ , but I can't find an example of sequence of functions $x_n$ that $\phi[x_n] \to 0$.

Answer 1

Hint: $$\int_0^1 e^{x(t)} x'(t)^2 dt \geq \left|\int_0^1 e^{x(t)/2}x'(t) dt\right|^2= |2e^{x(1)/2}-2e^{x(0)/2}|^2 = 4 (\sqrt{e^{\log 4}} -1)^2=4 $$