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Question

Let f:RR be a function such that f(x)=ax+3sinx+4cosx. Then f(x) is invertible if

A
a[5,5]
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B
a(,5]
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C
a[4,)
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D
a(,5][5,)
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Solution

The correct option is D a(,5][5,)
f(x)=ax+3sinx+4cosx
f(x)=a+3cosx4sinx
=a+5cos(x+α), where cosα=35
For invertible function, f(x) must be monotonic and onto
f(x)0 x or f(x)0 x
a+5cos(x+α)0 or a+5cos(x+α)0
a5cos(x+α) or a5cos(x+α)
a5 or a5
a(,5][5,)
Also f(x) is onto for x0.

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