If the function \(f:[-1,1] \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{cc}2^x+1, & \text { for }…
If the function \(f:[-1,1] \rightarrow R\) defined by \(f(x)=\left\{\begin{array}{cc}2^x+1, & \text { for } x \in[-1,0) \\ 1, & \text { for } x=0 \\ 2^x-1, & \text { for } x \in(0,1]\end{array}\right.\) then, in \([-1,1], f(x)\) has
a maximum
a minimum
both maximum and minimum
neither maximum nor minimum
Solution
The given function \(f:[-1,1] \rightarrow R\)
defined by \(f(x)=\left[\begin{array}{cc}2^x+1, & \text { for } x \in[-1,0) \\ 1, & \text { for } x=0 \\ 2^x-1, & \text { for } x \in(0,1]\end{array}\right.\)
The given function \(f(x)\) in \(x \in(-1,0)\) strictly increasing and \(f\left(0^{-}\right) \rightarrow 2\)
but \(f(0)=1\)
and in interval \(x \in(0,1)\), again it is strictly increasing, but \(f\left(0^{+}\right)=0\)
So, function has neither maximum nor minimum. Hence, option (4) is correct.