- #1
Stevela
- 2
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Homework Statement
Hello,
Suppose that f and g are differentiable functions satisfying
##\displaystyle \int_{0}^{f(x)} (fg)(t) \, \mathrm{d}t=g(f(x))##
Prove that g(0)=0
now if f(x)=0 in some point then it's straigh forward that g(f(x))=g(0)=0 anyways:
differentiating the first formula we get the following equation :
f'(x).(fg)(f(x))=g'(f(x)).f'(x)
let's suppose that f'(x)=0 , thus f is constant i.e f(x)=c, if c=0 we are done , g(0)=0 , if c=/=0 then :
##\displaystyle \int_{0}^{c} fg(t) \, \mathrm{d}t##=g(c)
##\displaystyle \int_{0}^{c} g(t) \, \mathrm{d}t##=g(c)/c, **i'm stuck here** , how can we prove that g(0)=0 (or get a contradiction) from this equation?