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)

Solution:

Maxima and minima are known as the extrema of a function.

Maxima and minima are the maximum or the minimum value of a function within the given set of ranges.

(i) The given function is f (x) = |x + 2| - 1

It can be observed that x + 2 ≥ 0 for every x ∈ R.

Therefore,

f (x) = x + 2 -1 ≥ - 1 for every x ∈ R .

The minimum value of f is attained when

|x + 2| = 0

⇒ x = - 2

x + 2 = 0

Therefore, Minimum value of

f = f (-2)

= |- 2 + 1| - 1

= - 1

Hence, the function f does not have a maximum value.

(ii) The given function is g (x) = - |x + 1| + 3

It can be observed that - |x + 1| ≤ 0 for every x ∈ R.

Therefore,

f (x) = - x + 1 + 3 ≤ 3 for every x ∈ R.

The maximum value of g is attained when |x + 1| = 0

|x + 1| = 0

⇒ x = - 1

Therefore, maximum value of

g = g (- 1)

= - |- 1 + 1| + 3

= 3

Hence, the function g does not have a minimum value.

(iii) The given function is h (x) = sin (2x) + 5

We know that Therefore, - 1 ≤ sin 2x ≤ 1

⇒ - 1+ 5 ≤ sin 2x ≤ 1 + 5

⇒ 4 ≤ sin 2x + 5 ≤ 6

Hence, the maximum and minimum values of h are 6 and 4, respectively.

(iv) The given function is f (x) = |sin 4x + 3|

We know that - 1 ≤ sin 4x ≤ 1

Therefore,

⇒ 2 ≤ sin 4x + 3 ≤ 4

⇒ 2 ≤ |sin 4x + 3| ≤ 4

Hence, the maximum and minimum values of f are 4 and 2, respectively.

(v) The given function is h (x) = x + 1, x ∈ (- 1, 1)

Here, if a point x₀ is closest to - 1, then we find x₀/2 + 1 > x₀ + 1 for all x₀ ∈ (- 1, 1).

Also, if x₁ is closest to 1, then find(x₁ + 1) < (x₁ + 1)/2 + 1 for all x₁ ∈ (- 1, 1).

Hence, function h (x) has neither maximum nor minimum value in (- 1, 1)

---

NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 2

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)

Summary:

i) f does not have a maximum value. ii) function g does not have a minimum value iii) the maximum and minimum values of h are 6 and 4 iv) the maximum and minimum values of f are 4 and 2 v) function h (x) has neither maximum nor minimum value in (- 1, 1)