Question

Find the maximum and minimum values, if any, of the following functions given by

(i) f (x) = |x + 2| – 1

(ii) g(x) = – | x + 1| + 3

(iii) h(x) = sin (2x) + 5

(iv) f (x) = |sin 4x + 3|

(v) h(x) = x + 1, x ∈ (- 1, 1)

Answer 1

(i) f(x) = |x + 2| – 1 

f(x) = |x + 2| – 1 ≥ -1 

minimum value is – 1 when x + 2 = 0, x = -2 

however it has no maximum value. 

(ii) g(x) = – | x + 1| + 3 = 3,-1 x + 1| 

g(x) < 3 v x + 1 = 0 

∴ max. value is 3 

when x = – 1 

how ever no minimum value. 

(iii)  h(x) = sin (2x) + 5

maximum value of sin 2x = 1 and minimum value is -1 

f(x) = 1 + 5 is 1 + 5 

∴ max. h (x) = 6 and 

min. h (x) = 4. 

(iv) f(x) = |sin 4x + 3| 

-1 < sin 4x < 1 ⇒ 3 -1 < sin 4x + 3 < + 1 + 3 + 2′< sin 4 x + 3 < 4 

2 < | sin 4x + 3 | < 4 

f(x) > 2 and f (x) < 4 

min. f(x) = 2 when sin 4x + 3 = 0 

max f (x) = 4 when sin 4x + 3 = 0 

∴ minimum value is 2 at sin 4x = -1 

maximum value is 4 at sin 4x = 1 

(v) h(x) = x + 1, x ∈ (-1,1) 

given that x ∈ (-1, 1) 

i.e. -1 < x < 1 

-1 + 1< x + 1 < 1 + 10 < x + 1< 2 

∴ x + 1 > 0 or x + 1 < 2 

x + 1 > 0 so no minimum value 

x + 1 < 2, so no maximum value.