Deriving an inequality for a set of functions that have equi-Lipschitz first derivatives.

Question

let $ F =\{ F_t \}_{t \in \mathbb{N}}$ be a sequence of continuously differentiable real valued functions such that the sequence $ F' =\{ F'_t \}_{t \in \mathbb{N}}$ is equi-Lipschitz , i.e. there exist a real constant $L$ such that for all $x_1,x_2$

$$ |f(x_1) - f(x_2)| \le L | x_1 - x_2 | $$

for all $f \in F'$.

Then is it true that $$F_{t+1} (x_1) - F_t(x_2) \le F'_t(x_1) (x_2 -x_1) + \frac{1}{2} L ( x_2 - x_1)^2 $$ for all $t$ ?

Answer 1

No. Let $F_t(x)=t$ for all $x\in\Bbb R$. Then $F'_t\equiv0$ fot all $t\in\Bbb N$ and $L=0$, but $F_{t+1}(x_1)-F_t(x_2)=1$ for all $t\in\Bbb N$ and all $x_i,x_2\in\Bbb R$.