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Question

Statement 1: If f(x) is differentiable in [0,1] such that f(0) =f(1)=0, then for any λR, there exists c such that f(c)=λf(c),0<c<1
Statement 2: If g(x) is differentiable in [0,1], where g(0) =g(1), then there exists c such that g(c)=0,0<c<1

A
Only statement 1 is true
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B
Only statement 2 is true
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C
Both are true
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D
Both are false
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Solution

The correct option is C Both are true
Statement 1 :
Consider, F(x)=eλxf(x),λR F(0)=f(0)=0F(1)=e1f(1)=0
By Rolle's theorem, F(c)=0 F(x)=eλx(f(x)λf(x))F(c)=0eλc(f(c)λf(c))=0f(c)=λf(c),0<c<1

Statement 2 :
By rolle's theorem, second statement is true.

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