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) = 4x2 – 4x + 4 on R
Answer» Given as f(x) = 4x² – 4x + 4 on R = 4x² – 4x + 1 + 3 On grouping the above equation we get, = (2x – 1)² + 3 Here, (2x – 1)² ≥ 0 = (2x – 1)² + 3 ≥ 3 = f(x) ≥ f (1/2) Hence, the minimum value of f(x) is 3 at x = 1/2 Clearly, f(x) can be made large. So maximum value does not exist.

Find the maximum and the minimum values, if any, without using derivatives of the functions:f (x) = 4x2 – 4x + 4 on R

Answer»

Given as f(x) = 4x² – 4x + 4 on R

= 4x² – 4x + 1 + 3

On grouping the above equation we get,

= (2x – 1)² + 3

Here, (2x – 1)² ≥ 0

= (2x – 1)² + 3 ≥ 3

= f(x) ≥ f (1/2)

Hence, the minimum value of f(x) is 3 at x = 1/2

Clearly, f(x) can be made large. So maximum value does not exist.