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Question

Consider a differential function f(x) on the set of real numbers such that f(1)=0 and |f(x)|2. Given these conditions, which one of the following inequalties is necessarily ture for all xϵ[2,2]?

A
f(x)2|x+1|
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B
f(x)12|x+1|
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C
f(x)2|x|
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D
f(x)12|x|
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Solution

The correct option is A f(x)2|x+1|
Conditions given
f(1)=0 and |f'(x)|\leq 2\)
Consider option (1)
f(x)=2|x+1|



f(1)=0, satisfied
f(x)|2, satisfied
Consider opiton (2)
f(x)=12|x+1|


f(1)=0, satisfied
If |f(x)|2, not satisifed
Consider option (3)



f(1)=0, satisfied
It |f(x)2, satisfied
consider option (4)
f(x)=12|x|



f(1)=0, not satisfied
|f(x)2, not satisfied.

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