Combination of function values on subinterval is bounded by $\inf$/$\sup$ of function.

Question

Let $f:[a,b] \to \mathbb{R}$ be a real valued function. And $\mathcal{P} = (x_0, \ldots, x_n)$ be a partition of $[a,b]$. Why is $$\inf_{x\in[x_{k-1},x_{k}]}f(x) \leq f(x_{k-1})+\dfrac{f(x_k) - f(x_{k-1})}{x_k-x_{k-1}}(x-x_{k-1}) \leq \sup_{x\in[x_{k-1},x_{k}]}f(x)$$ for all $x\in [x_{k-1}, x_k)$ and $k=1,\ldots,n$