J[y] = \(\displaystyle \int_0^1{e^y(y')^2 dx\)

y(0) = 1, y(1) = 2ln(2).

I need to find the extremal and show whether it provides a maximum or minimum.

In this case I only have problems with the second part. I have the extremal as y(x) = 2ln(x(2-\(\displaystyle \sqrt{e}\))+\(\displaystyle \sqrt{e}\)).

To find whether its a maximum or minimum the process I follow is; consider v(x) such that v(0)=0 and v(1)=0, and find the sign of:

J[y+v] - J[y] = \(\displaystyle \int_0^1{e^{y+v}((y+v)')^2 dx\) - \(\displaystyle \int_0^1{e^y(y')^2 dx\).

I have that this should come out as positive in my notes and thus y provides a minimum - but I don't know how to get there. Squaring out the J[y+v] integral bracket gives me terms I don't know how to integrate, and integration by parts has proved unable to alleviate this problem. I fear I may be going wrong with my approach. Does all of what I've written make sense? Can someone offer any pointers?

Thanks!