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Justification with the mean value theorem


Let f(x)=2xsin(πx).
Below is Rafael's attempt to write a formal justification for the fact that the equation f(x)=14 has a solution where 2<x<1.
Is Rafael's justification complete? If not, why?
Rafael's justification:
Exponential and trigonometric functions are differentiable and continuous at all points in their domain, and 2x1 is within f's domain.
So, according to the mean value theorem, f(x)=14 must have a solution somewhere in the interval 2<x<1.
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