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Question

f(x)=x3+3x2+4x+bsinx+ccosx,xR is a one-one function, then the value of b2+c2 is

A
1
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B
2
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C
1
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D
none of these
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Solution

The correct option is D 1
Here, f(x)=x3+3x2+4x+bsinx+ccosx
f(x)=3x2+6x+4+bcosxcsinx
Now, for f(x) to be one-one, the only possibility is
f(x)0,xR
ie, 3x2+6x+4+bcosxcsinx0,xR
ie, 3x2+6x+4csinxbcosx,xR
As we are approximating lower bound for the function by a number instead of a function, hence we chose the number.
ie, 3x2+6x+4b2+c2,xR
ie, b2+c23(x2+2x+1)+1,xR
b2+c23(x+1)2+1,xR
b2+c21,xR
b2+c21,xR
Hence, (c) is the correct answer.

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