Find the maximum and the minimum values, if any, of the function given by $f(x) = x, x \in (0, 1).$

Neither Maximum nor Minimum exists

The correct answer is Option (4) → Neither Maximum nor Minimum exists ##

The given function is an increasing (strictly) function in the given interval $(0, 1)$. From the graph of the function $f$, it seems that, it should have the minimum value at a point closest to $0$ on its right and the maximum value at a point closest to $1$ on its left. Are such points available? Of course, not. It is not possible to locate such points. Infact, if a point $x_0$ is closest to $0$, then we find $\frac{x_0}{2} < x_0$ for all $x_0 \in (0, 1)$. Also, if $x_1$ is closest to $1$, then $\frac{x_1 + 1}{2} > x_1$ for all $x_1 \in (0, 1)$.

Therefore, the given function has neither the maximum value nor the minimum value in the interval $(0, 1)$.