Let $a$ be a real number satisfying $a \geqq 0$. We are to express the maximum value $M$ of the function $f(x) = |x^2 - 2x|$ on the range $a \leqq x \leqq a + 1$ in terms of $a$. Furthermore, we are to find the minimum value of $M$ over the range $a \geqq 0$. (1) The function $f(x)$ can be expressed without using the absolute value symbol as follows: when $x \leqq \mathbf{M}$ or $x \geqq \mathbf{N}$, then $f(x) = x^2 - 2x$; when $\mathbf{M} < x < \mathbf{N}$, then $f(x) = -x^2 + 2x$. Hence, the maximum value of $f(x)$ on $a \leqq x \leqq a + 1$ is the following: when $0 \leqq a \leqq \mathbf{O}$, then $M = \mathbf{P}$; when $\mathbf{O} < a \leqq \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = -a^2 + \frac{\mathbf{T}}{\mathbf{S}}a$; when $a > \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = a^2 - \mathbf{U}$. (2) The minimum value of $M$ over the range $a \geqq 0$ is $\frac{\sqrt{\mathbf{V}}}{\mathbf{W}}$.
Let $a$ be a real number satisfying $a \geqq 0$. We are to express the maximum value $M$ of the function $f(x) = |x^2 - 2x|$ on the range $a \leqq x \leqq a + 1$ in terms of $a$. Furthermore, we are to find the minimum value of $M$ over the range $a \geqq 0$.
(1) The function $f(x)$ can be expressed without using the absolute value symbol as follows: when $x \leqq \mathbf{M}$ or $x \geqq \mathbf{N}$, then $f(x) = x^2 - 2x$; when $\mathbf{M} < x < \mathbf{N}$, then $f(x) = -x^2 + 2x$.
Hence, the maximum value of $f(x)$ on $a \leqq x \leqq a + 1$ is the following:
when $0 \leqq a \leqq \mathbf{O}$, then $M = \mathbf{P}$;
when $\mathbf{O} < a \leqq \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = -a^2 + \frac{\mathbf{T}}{\mathbf{S}}a$;
when $a > \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = a^2 - \mathbf{U}$.
(2) The minimum value of $M$ over the range $a \geqq 0$ is $\frac{\sqrt{\mathbf{V}}}{\mathbf{W}}$.