Find the maximum and the minimum values, if any, without using derivatives of the following functions:
f(x) = 4x2 – 4x + 4 on R
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
Thus, the minimum value of f(x) is 3 at x = 
Since, f(x) can be made large. Therefore maximum value does not exist.
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