Find the maximum and the minimum values, if any, without using derivatives of the following functions:
f(x) = –(x – 1)2 + 2 on R
We have f(x) = – (x – 1)2 + 2
It can be observed that (x – 1)2≥0 for every x∈R
Therefore, f(x) = – (x – 1)2 + 2≤2 for every x∈R
The maximum value of f is attained when (x – 1) = 0
(x – 1)=0, x=1
Since, Maximum value of f = f(1) = – (1 – 1)2 + 2 = 2
Hence, function f does not have minimum value.