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Question

If f(x),g(x) be twice differential functions on [0,2] satisfying f′′(x)=g′′(x), f(1)=2g(1)=4 and f(2)=3g(2)=9, then

A
f(4)g(4)=10
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B
|f(x)g(x)|<22<x<0
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C
f(2)=g(2)x=1
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D
f(x)g(x)=2x has real root
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Solution

The correct option is C f(2)=g(2)x=1

We have f′′(x)=g′′(x). On integration, we get
f(x)=g(x)+C ...(1)
Putting x=1, we get
f(1)=g(1)+C or 4=2+C or C=2
f(x)=g(x)+2

Integrating w.r.t. x, we get f(x)=g(x)+2x+c1 (2)
Putting x=2, we get
f(2)=g(2)+4+c1 or 9=3+4+c1 or c1=2
f(x)=g(x)+2x+2
Putting x=4, we get f(4)g(4)=10
|f(x)g(x)|<2 or |2x+2|<2 or |x+1|<1 or 2<x<0
Also, f(2)=g(2) or x=1
f(x)g(x)=2x has no solution.

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