Euler–Lagrange equation has no solutions

Show that the Euler–Lagrange equation for the functional:

$$I(y) = \int_{0}^{1}y dx$$

subject to y(0) = y(1) = 0 has no solutions. Explain why no extremum for I exists.

When forming the E-L equation I get 1=0. How would I go about doing this question?

If you consider the function $$f_n$$, where $$f_n(x) = n, x \in [\frac{1}{n}, 1 -\frac{1}{n}]$$, $$f_n(x) = n^2x, x\in [0, \frac{1}{n}],$$ and $$f_n(x) = n^2(1-x), x\in [1 -\frac{1}{n}, 1],$$ then clearly $$I(f_n)$$ can be as large as you want. $$I(-f_n)$$ can also be as large as you want.