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if f'(x)>0, then is it necessary for f(x) to be one one? ALSO, f(x)=2x+sin x .please tell if it's one-one or onto. please reply as soon as possible.......

if f'(x)>0, then is it necessary for  f(x) to be one one?


ALSO,


f(x)=2x+sin x .please tell if it's one-one or onto.


please reply as soon as possible.......


 


 


 

Grade:12

1 Answers

mycroft holmes
272 Points
10 years ago

If x1 /= x2 , then by Lagrange's Mean Value Theorem, we have f(x2) - f(x1)/(x2-x1) = f'(c) for some c in the interval (x1 , x2).

 

Since f'(c)>0, and x2 > x1 , this implies f(x2) > f(x1). Thus, we have that if x1 /= x2 , then f(x2) /= f(x1).

 

Thus f'(x)>0 proves that f(x) is one-one

 

In general if f(x) is monotonic, it is also one-one.

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