Justification with the mean value theorem

Let

f (x) = 2^{x} - sin (π x)

Below is Rafael's attempt to write a formal justification for the fact that the equation

f^{'} (x) = \frac{1}{4}

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 $- 2 \leq x \leq - 1$ ‍ is within $f$ ‍'s domain.
So, according to the mean value theorem, $f^{'} (x) = \frac{1}{4}$ ‍ must have a solution somewhere in the interval $- 2 < x < - 1$ ‍.

Διαφορικός Λογισμός

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