It is known that given $ a, \ b \in \mathbb{R}^n $, $ v \in \mathbb{R}^m $ and a differentiable function $ f : \mathbb{R}^n \rightarrow \mathbb{R}^m $, the following holds
$$ v \cdot (f(b) - f(a)) = v \cdot Df(c)(b - a) $$
for some point $ c \in [a, b] = \mbox{segment joining points } a \mbox{ and } b \mbox{ in } \mathbb{R}^n $. As a consequence we derive the following inequality:
\begin{equation} \label{Ineq} || f(b) - f(a) || \leq || Df(c)(b - a) || \end{equation}
Now I have been asked about how to prove this inequality is not an equality in general. What I understand is that I should find a function $ f $ and choose $ a $ and $ b $ such that the inequality is strict for any $ c \in [a, b] $. But I have tried with some easy functions and haven't founded it. Can someone give me some hint about this?
Take $f:\mathbb{R} \to \mathbb{R}^2$, $f(x)=(\cos(x),\sin(x))$ with $a=0,b=2\pi$. Then $\|f(b)-f(a)\|=0$ but for all $c \in \mathbb{R}$ $$ \|Df(c)(b-a)\| = 2\pi \|(-\sin(c),\cos(c))\| = 2\pi. $$