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Question

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

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Solution

f(x)=x3+3x2+4x+bsinx+ccosx
Differentiating w.r.t. x, we get
f(x)=3x2+6x+4+bcosxcsinx
For f(x) to be one-one, only possibility is f(x)0 xR

3x2+6x+4+bcosxcsinx0 for all xR
3x2+6x+4csinxbcosx for all xR
3x2+6x+4b2+c2
b2+c23(x2+2x+1)+1
b2+c23(x+1)2+1
b2+c21
b2+c21

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