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 following functions :f(x) = 4x2 – 4x + 4 on R
Answer» f(x) = 4x²– 4x + 4 on R = 4x²– 4x + 1 + 3 = (2x – 1)² + 3 Since, (2x – 1)² ≥ 0 = (2x – 1)² + 3 ≥ 3 = f(x) ≥ f(\(\frac{1}{2}\)) Thus, the minimum value of f(x) is 3 at x = \(\frac{1}{2}\) Since, f(x) can be made large. Therefore maximum value does not exist.

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

Answer»

f(x) = 4x²– 4x + 4 on R

= 4x²– 4x + 1 + 3

= (2x – 1)² + 3

Since,

(2x – 1)² ≥ 0

= (2x – 1)² + 3 ≥ 3

= f(x) ≥ f(\(\frac{1}{2}\))

Thus, the minimum value of f(x) is 3 at x = \(\frac{1}{2}\)

Since,

f(x) can be made large.

Therefore maximum value does not exist.