According to Lagrange's Mean Value Theorem (LMVT), if a function $f(x)$ is continuous on $\left[a,b\right]$ and differentiable on $\left(a,b\right)$, then there exists some constant $c$ such that $$f'(c) = \frac{f(b) - f(a)}{b-a} $$ Now, I have the following question: Q) Using Lagrange's Me...

Let $a<b$ be positive real numbers , then is it true that there exist $x,y \in (a,b)$ such that $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$ ?

A search result for Mean Value Theorem gives us 2715 results, and results on the page are like ones I think we can include in the tag. The theorem is an important result in calculus, and questions relating to its applications, proofs. I think it would be useful if could have the tag, as it can gr...

It seems like there are a decent number of questions about placing some sort of order on a group. I would like to tag these questions to better organize them, but I'm not sure what to call the tag. The phrase "ordered group" to me implies that we have a total order $\leq$ which is preserved unde...