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∫wvdx=0 (int from 0 to 1) for all v in V. w is continuous on [0,1]. What it means to be in V: v in V satisfies being continuous on [0,1], v(0)=v(1)=0, and derivatives of v are piecewise continuous .

Problem is:

Show that w(x)=0 for x in [0,1].

I have spent hours with this. The book I'm looking at describes a couple problems:

for u in V, u is a solution to -u''=f(x), u is also a solution to (u',v')=(f,v) and also a solution to

1/2(u',u')-(f,u) less than or equal to 1/2(v',v')-(f,v).

I have tried in many combinations to use this information, and this is the only given info in the chapter.

Help.