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Question

If f:RR+ is a function defined as [f(x)]2=x0[{f(t)}2+{f(t)}2]dt+100, then f(ln2)=  


Solution

[f(x)]2=x0[{f(t)}2+{f(t)}2]dt+100
Differentiating w.r.t x
2f(x).f(x)=[f(x)]2+[f(x)]2[ddxxaf(t)dt=f(x)]
[f(x)f(x)]2=0f(x)=f(x)f(x)f(x) dx= dx+cln|f(x)|=x+lnCf(x)=Cex
Now, for x=0
[f(0)]2=00[{f(0)}2+{f(0)}2]dt+100[f(0)]2=100f(0)=10  (f:RR+)
f(x)=10ex

 

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