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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