Taylor formula (assuming $ x_0 = 0$ ) for function $ f$ with Lagrange remainder is $ f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f”(0)}{2!}x^2 + … + \frac{f^{(n)}(0)}{n!}x^n + \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$ , where $ c \in (0, x)$ if $ x > 0$ and $ c \in (x, 0)$ if $ x < 0$ .

Is there any characterization of $ c$ ? Can we tell anything about what $ c$ might be or might not be, except that it’s in this interval? Sometimes when solving problems I feel like I could simplify my solutions if I’d known something more about this $ c$ .