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Question

If c is a point at which Rolle's theorem holds for the function, f(x)=loge(x2+α7x) in the interval [3,4], where aR, then f"(c) is equal to:

A
124
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B
112
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C
37
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D
112
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Solution

The correct option is D 112
Rolle's theorem is applicable on f(x) in [3,4]
f(3)=f(4)
ln(9+α21)=ln(16+α28)
9+α21=16+α28
36+4α=48+3αα=12

Now, f(x)=ln(x2+127x) f(x)=7xx2+12×(17127x2)
f(x)=x212x(x2+12)

f(c)=0c=23

f′′(x)=x4+48x2+144x2(x2+12)2
f′′(c)=f′′(23)=112

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