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

(i) f (x) = (2x – 1)2 + 3 

For all values of x, f (x) > 3 (2x – 1)² + 3 > 3 

∴ minimum value is 3 when 2x -1 = 0 is x = \(\frac{1}{2}\)however the function has no maximum values as f (x) → ∞ as |x| ∞

(ii) f(x) = 9x² + 12x + 2

However there is no maximum values.

(iii) f(x) = – (x – 1)² + 10 

f(x)= 10 – (x – 1)² 

10 – (x – 1)² < 10 

f(x) has a maximum value when x – 1 = 0,x= 1. 

how ever f (x) has no minimum value. 

(iv) g(x) = x³ + 1 as  → ∞ g (x) → ∞ and a s → ∞, g(x) → – ∞ 

∴ g(x) has neither minimum value nor maximum value.