Let's understand one-one function and differentiability with the help of a problem. Try it yourself before reading the solution.
Let f be real valued, differentiable on (a, b) and $ f'(x) \ne 0 $ for all $ x \in (a, b) $. Then f is 1-1.
True
Discussion:
Suppose f is not 1-1. Then there exists $x_1 , x_2 \in (a, b) $ such that $ f(x_1 ) = f(x_2) $. Since f(x) is differentiable it must be continuous as well. Applying Rolle's Theorem in the interval $ (x_1 , x_2 ) $ we conclude that there exists a number c in this interval such that f'(c) = 0. But this contradicts the given conditions. Hence f must be 1-1
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