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anjali agarwal Grade: 12
        

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.......


 


 


 

7 years ago

Answers : (1)

mycroft holmes
266 Points
										

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.

7 years ago
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