Let the function $f : [0,1] \rightarrow \mathbb{R}$ be defined as $$f(x) = \max\left\{\frac{|x - y|}{x + y + 1} : 0 \leq y \leq 1\right\} \text{ for } 0 \leq x \leq 1$$ Then which of the following statements is correct? (A) $f$ is strictly increasing on $\left[0, \frac{1}{2}\right]$ and strictly decreasing on $\left[\frac{1}{2}, 1\right]$. (B) $f$ is strictly decreasing on $\left[0, \frac{1}{2}\right]$ and strictly increasing on $\left[\frac{1}{2}, 1\right]$. (C) $f$ is strictly increasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly decreasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$. (D) $f$ is strictly decreasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly increasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.
Let the function $f : [0,1] \rightarrow \mathbb{R}$ be defined as
$$f(x) = \max\left\{\frac{|x - y|}{x + y + 1} : 0 \leq y \leq 1\right\} \text{ for } 0 \leq x \leq 1$$
Then which of the following statements is correct?\\
(A) $f$ is strictly increasing on $\left[0, \frac{1}{2}\right]$ and strictly decreasing on $\left[\frac{1}{2}, 1\right]$.\\
(B) $f$ is strictly decreasing on $\left[0, \frac{1}{2}\right]$ and strictly increasing on $\left[\frac{1}{2}, 1\right]$.\\
(C) $f$ is strictly increasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly decreasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.\\
(D) $f$ is strictly decreasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly increasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.