Find the maximum or minimum value, if any, without using derivatives,

1.	Find the maximum or minimum value, if any, without using derivatives, of the function: \|sin 4x + 3\|
Answer» For all x – 1 ≤ sin 4x ≤ 1 By adding 3 3 – 1 ≤ sin 4x + 3 ≤ 1 + 3 On further calculation 2 ≤ sin 4x + 3 ≤ 4 We know that \|2\| ≤ \|sin 4x + 3\| ≤ \|4\| 2 ≤ f (x) ≤ 4 Here the minimum value of f (x) = 2 and maximum value of f (x) = 4.

Find the maximum or minimum value, if any, without using derivatives, of the function: |sin 4x + 3|

Answer»

For all x

– 1 ≤ sin 4x ≤ 1

By adding 3

3 – 1 ≤ sin 4x + 3 ≤ 1 + 3

On further calculation

2 ≤ sin 4x + 3 ≤ 4

We know that

|2| ≤ |sin 4x + 3| ≤ |4|

2 ≤ f (x) ≤ 4

Here the minimum value of f (x) = 2 and maximum value of f (x) = 4.