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Question

If f:RR is a twice differentiable function such that f′′(x)>0 for all xR, and f(12)=12, f(1)=1, then

A
f(1)0
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B
0<f(1)12
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C
12<f(1)1
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D
f(1)>1
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Solution

The correct option is D f(1)>1
f(12)=12, f(1)=1Let g(x)=f(x)x, x[12,1]g(12)=g(1)=0
Using Rolle's theorem, we get
g(c)=0, c(12,1)f(c)1=0 [g(x)=f(x)1]f(c)=1, c(12,1)As f′′(x)>0,f(x) is increasing on Rf(1)>f(c)f(1)>1

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