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 + 4, x ∈(−1, 1)

Answer 1

(i)f (x) = |x + 2| – 1
Now, we know that |x + 2| ≥ 0 for every x ∈ R
⇒ f (x) = |x + 2| – 1 ≥ -1 for every x ∈ R
hence, minimum value of f(x) is -1
you can also solve by 2nd method.
The minimum value of f(x) is attained when |x + 2| = 0
e.g., |x + 2| =0
⇒ x = -2
Then, Minimum value of f(x) = f(-2)
= |-2 + 2| - 1 = -1
Therefore, function f does not have a maximum value.

(ii) g(x) = –|x + 1| + 3
Now, we can see that –|x + 1| ≤ 0 for every x ∈ R
⇒ g(x) = –|x + 1| + 3 ≤ 3 for every x ∈ R.
hence , the maximum value of f(x) = 3

you can also solve by 2nd method:
The maximum value of f is attained when |x + 1| = 0
e.g., |x + 1| = 0
⇒ x = -1
Then, Maximum value of g(x) = g(-1)
= -|-1 + 1| + 3 = 3
Therefore, function g(x) does not have a minimum value.

(iii) h(x) = sin(2x) + 5
Now, we know , -1 ≤ sin2x ≤ 1
⇒ -1 + 5 ≤ sin2x + 5 ≤ 1 + 5
⇒ 4 ≤ sin2x + 5 ≤ 6
Therefore, the maximum and minimum value of function h are 6 and 4 respectively.


(iv) f(x) = |sin 4x + 3|
Now, we know, -1 ≤ sin4x ≤ 1
⇒ 2 ≤ sin 4x + 3 ≤ 4
⇒ 2 ≤ |sin 4x + 3| ≤ 4
Therefore, the maximum and minimum value of function h are 4 and 2 respectively.

(v) h(x) = x + 1 , x∈ (-1,1)
differentiate with respect to x ,
h'(x) = 1 ≠ 0
hence, function h(x) has neither maximum value nor minimum value in (-1,1)