Euler Lagrange equation for finding extremal value for $\int_0^1 xw'(x)dx$ given $w(0)=0,w(1)=10$

Question

Euler Lagrange equation for finding extremal value for $\int_0^1 xw'(x)dx$ given $w(0)=0,w(1)=10$

Based on Euler-Lagrange(EL) equation, we have

\begin{align} \frac{\partial xw'(x)}{\partial w(x)}-\frac{d}{dx}(\frac{\partial xw'(x)}{\partial w'(x)})&=0\\ \implies 0-\frac{d}{dx}(x)&=0 \end{align} However, the above equation does not hold. Since EL equation fails to hold, does it imply that $\int_0^1 xw'(x)dx$ does not have minimum or maximum value for any choice of $w$? In another words, integral can be $\infty$ or $-\infty$ .

Answer 1

The fact that your EL equation can never hold means that there are no critical values for the integral, so the integral is unbounded. Consider the function

$$w(x)=ax^2+(10-a)x$$

which satisfies the conditions you have described. We then have

$$xw'(x)=2ax^2+(10-a)x\implies\int_0^1xw'(x)dx=-\frac{1}{6}a-5$$

We may take $a$ as large or as small as we would like to make the integral as large or as small as we would like.

I hope this helps!