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.......

Question

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.......

mycroft holmes · Answer

If x 1 /= x 2 , then by Lagrange's Mean Value Theorem, we have f(x 2 ) - f(x 1 )/(x 2 -x 1 ) = f'(c) for some c in the interval (x 1 , x 2 ). Since f'(c)>0, and x 2 > x 1 , this implies f(x 2 ) > f(x 1 ). Thus, we have that if x 1 /= x 2 , then f(x 2 ) /= f(x 1 ). 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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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.......

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