Question

the maximum value of sin x + cos x

Answer 1

hey mate here is your answer the maximum value of sin x + cos x will be 1

Answer 2

Suppose we define the given function as

y = f(x) = sin x + cos x
Then, differentiating w.r.t x, we get

dy/dx = cos x - sin x

Then, equating dy/dx to 0

cos x - sin x = 0
cos x = sin x
tan x = 1 or cot x = 1
Therefore,
x = 45° or π/4 rad.

Now
since we got the values for x

We find out the second derivative of y

d2y / dx2 = - sin x - cos x

Now, putting the value of x in d2y/dx2
We get,

(d2y/dx2) at 45° = - sin 45° - cos 45° = -1/√2 - 1/√2 = -√2

Since the value obtained is negative this confirms that the value here of y will be Maximum since we know that the value of a particular function is maximum when it's second derivative is negative for the value of x.

Hence,
The max value of sin x + cos x
= sin 45° + cos 45°
= 1/√2 + 1/√2 = 2/√2 = √2

Hence the max value of y is √2 when x is at 45° or π/4.