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

Let f(x) = sin x + cos x,

⇒ f’(x) = cosx - sinx

Now, f’(x) = 0

⇒ cosx - sinx = 0

⇒ cosx = sinx

⇒ tanx = 1

Now,

If f’’(x) will be negative when (sinx + cosx) > 0, means both sinx and cosx are positive.

And, we know that sinx and cosx both are positive in the first quadrant.

Then, f’’(x) will be negative when

f’’(x) = -sinx – cosx = -(sinx + cosx)

Now, let us take x =

Then, by second derivative test,

f will be maximum at x =

And, the maximum value of f is