Question

Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x − 1)2 + 3 (ii) f(x) = 9x2+12x+2 (iii) f(x) = −(x − 1)2+ 10 (iv) g(x) = x3 +1

Answer 1

The given function is f(x) = (2x − 1)2 + 3. It can be observed that (2x − 1)2 ≥ 0 for every x ∴ R. Therefore, f(x) = (2x − 1)2+3 ≥ 3 for every x ∴ R. The minimum value of f is attained when 2x − 1 = 0.

3x + 2 = 0 ∴ x=$-\frac{2}{3}$

∴The minimum value of f = f($-\frac{2}{3}$)=(3($-\frac{2}{3}$)+2)2

Hence, function f does not have a maximum value.

The given function is f(x) =−(x−1)2+10.

It can be observed that (x − 1) 2 ≥ 0 for every x ∴ R.