Find the minimum value taken by the following two integral expressions where and are both functions of

(a)

(b)

I've had a few stabs at this but my answers are looking dodgy. Could someone please show me the way? ^^

Here is what I've done so far:

(a) Using Euler Lagrange Equation , I get

so

so by integrating the last expression for I get

Applying boundary conditions again, I get

Substituting and rearranging, I end up with as the extremal funcation.

For part (b) I did something similar using as the Euler Langrage equation and ended up with

Here is my working out for part (b):

so using , I get

Taking square roots of both sides and multiplying the whole equation by -1, I get

Using boundary condition , I get

Substituting and rearranging, I get

Using the second boundary condition, I get

Hence,