Find the maximum and minimum values, if any, of the following function given by,

$h (x) = x + 1, x ϵ (- 1, 1)$

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Solution

Given functions is, $h (x) = x + 1, - 1 < x < 1$
Now $- 1 < x < 1 \Leftrightarrow - 1 + 1 < x + 1 < 1 + 1 \Leftrightarrow 0 < x + 1 < 2$
Hence, on the point (0,2), f has neither a maximum nor a minimum value.

Alternate method:
h'(x)=1 exists at all $x ϵ (- 1, 1)$
Also, $h^{'} (x) \neq 0$ for any $x ϵ (- 1, 1)$
Further h(x) is defined on an open interval.
So, there is no point for which f(x) may have a minimum or a maximum value.

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