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Question

Let f:[π2,π2]R be a continuous function such that f(0)=1 and π/30f(t)dt=0

Then which of the following statements is(are) TRUE ?

A
The equation f(x)3cos3x=0 has at least one solution in (0,π3)
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B
The equation f(x)3sin3x=6π has at least one solution in (0,π3)
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C
limx0xx0f(t)dt1ex2=1
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D
limx0sinxx0f(t)dtx2=1
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Solution

The correct option is C limx0xx0f(t)dt1ex2=1
f(0)=1 and π/30f(t)dt=0

Consider function g(x)=x0f(t)dtsin3x
g(x) is continuous and differentiable function.
g(0)=0 and g(π3)=0
By Rolle's theorem, g(x)=0 has at least one solution in (0,π3)
So, f(x)3cos3x=0 for some x(0,π3)


Consider function h(x)=x0f(t)dt+cos3x+6πx
h(x) is continuous and differentiable function and h(0)=1, h(π3)=1
By Rolle's theorem, h(x)=0 for at least one x(0,π3)
So, f(x)3sin3x+6π=0 for some x(0,π3)


L=limx0xx0f(t)dt1ex2 (00 form)
By L' Hopital rule,
L=limx0xf(x)+x0f(t)dt2xex2 (00 form)
Again by L' Hopital rule,
L=limx0xf(x)+f(x)+f(x)4x2ex22ex2=0+2f(0)02=1


limx0sinxx0f(t)dtx2 (00 form)=limx0 sinxf(x)+cosxx0f(t)dt2x (00 form)
=limx0cosxf(x)+sinxf(x)+cosxf(x)sinxx0f(t)dt2=1+0+102=1

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