Find the maximum and the minimum values, if any, without using derivat

1.	Find the maximum and the minimum values, if any, without using derivatives of the functions: f(x) = –(x - 1)2 + 2 on R
Answer» Given as f(x) = – (x – 1)² + 2 It can be observed that (x – 1)²≥ 0 for every x ∈ R So, 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 Here, Maximum value of f = f (1) = – (1 – 1)² + 2 = 2 Thus, function f does not have minimum value.

Find the maximum and the minimum values, if any, without using derivatives of the functions: f(x) = –(x - 1)2 + 2 on R

Answer»

Given as f(x) = – (x – 1)² + 2

It can be observed that (x – 1)²≥ 0 for every x ∈ R

So, 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

Here, Maximum value of f = f (1) = – (1 – 1)² + 2 = 2

Thus, function f does not have minimum value.