# Find the maximum and minimum values of x + sin 2x on [0, 2π]

**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

Let f (x) = x + sin 2x

Therefore,

f' (x) = 1 + 2 cos 2x

Now,

f' (x) = 0

⇒ 1 + 2 cos 2x = 0

⇒ cos 2x = - 1/2 = - cos π/3 = cos (π - π/3) = cos 2π/3

⇒ 2x = 2nπ ± 2π / 3 [n ∈ z]

⇒ x = nπ ± π / 3 [n ∈ z]

x = π / 3, 2π / 3, 4π / 3, 5π / 3 ∈ [0, 2π]

Then, we evaluate the value of f at critical points x = π/3, 2π/3, 4π/3, 5π/3 and at the end points of the interval [0, 2π].

f (π/3) = π/3 + sin 2(π / 3)

= π / 3 + √3 / 2

f (2π/3) = 2π/3 + sin 2(2π / 3)

= 2π / 3 - √3 / 2

f (4π/3) = 4π/3 + sin 2(4π / 3)

= 4π / 3 + √3 / 2

f (5π/3) = 5π/3 + sin 2(5π / 3)

= 5π / 3 - √3 / 2

f (0) = 0 + sin 0

= 0

f (2π) = 2π + sin 4π

= 2π + 0

= 2π

Hence, we can conclude that the absolute maximum value of f (x) in the interval [0, 2π] is 2π occurring at x = 2π and the absolute minimum value of f (x) in the interval [0, 2π] is 0 occurring at x = 0

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

## Find the maximum and minimum values of x + sin 2x on [0, 2π].

**Summary:**

The absolute maximum value of f (x) in the interval [0, 2π] is 2π occurring at x = 2π and he absolute minimum value of f (x) in the interval [0, 2π] is 0 occurring at x = 0

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