What is the maximum value of the function sin x + cos x?

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) = sin x + cos x

Therefore,

On differentiating wrt x, we get

f' (x) = cos x - sin x

Now,

f' (x) = 0

⇒ cos x - sin x = 0

⇒ sin x = cos x

On dividing both sides by cos x, we get

⇒ tan x = 1

⇒ x = π / 4, 5π / 4, ...

Hence,

On further differentiating,

f" (x) = - sin x - cos x

= - (sin x + cos x)

Now, f" (x) will be negative when (sin x + cos x) is positive i.e., when sin x and cos x are both positive.

Also, we know that sin x and cos x both are positive in the first quadrant.

Then, f" (x) will be negative when x ∈ (, π / 2)

Thus, we consider x = π / 4

f" (π/4) = - sin (π/4) - cos (π/4)

= (- 2/√2)

= - √2 < 0

By the second derivative test, f will be the maximum at x = π/4 and the maximum value of f is

f (π/4) = sin (π/4) + cos (π/4)

= 1/√2 + 1/√2

= 2/√2

= √2

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.5 Question 9

What is the maximum value of the function sin x + cos x?

Summary:

The maximum value of the function sin x + cos x is √2. Maxima and minima are the maximum or the minimum value of a function within the given set of ranges