Question
Find the maximum and the minimum values, if any, of the function given by $f(x) = x, x \in (0, 1)$.
✓
Answer
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. It is not possible to locate such points.
In fact, if a point $x_0$_ is closest to $0,$ then we find $\frac{x_{0}}{2} < x$ 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 $(0,1)$.
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. It is not possible to locate such points.
In fact, if a point $x_0$_ is closest to $0,$ then we find $\frac{x_{0}}{2} < x$ 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 $(0,1)$.
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