In H. Cartan's Differential Calculus, Theorem 3.1.1 is called the Mean Value Theorem and is stated as:
Theorem: Let $f:[a,b]\to\mathbf R^n$ and $g:[a,b]\to\mathbf R$ be two functions which are continuous on $[a,b]$ and differentiable on $(a,b)$. Assume that $\|f'(x)\|\leq g'(x)$ for all $x\in(a,b)$. Then $\|f(b)-f(a)\|\leq g(b)-g(a)$.
On the other hand, the universally known (Lagrange's) Mean Value Theorem states that
Theorem: Let $f:[a,b]\to\mathbf R$ be a function which is continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $c\in(a,b)$ such that $(f(b)-f(a))/(b-a)=f'(c)$.
Since they are both being called Mean Value Theorems, I think may be one follows easily from the other.
But I fail to see the connection.
Can somebody please shed some light on this.