Question

Show that the function f given by f(x) = sin x + cos x, is strictly decreasing in the interval `(pi/4,(5pi)/4)`.

Answer 1

Given f(x) = sin x + cos x

f'(x) = cos x − sin x

putting f'(x) = 0

cos x = sin x

`x = pi/4,(5pi)/4`   ...(for x ∈ [0, 2π])

plotting points

Here, when `x ∈ pi/4, (5pi)/4`

putting f'(x) = cos x − sin x

at `x = pi/2 ∈ (pi/4, (5pi)/4)`

`f' (pi/2) = cos  pi/2 - sin  pi/2=-1<0`

thus f'(x) < 0 for x ∈ `(pi/4, (5pi)/4)`

⇒ f is strictly decreasing in x ∈ `(pi/4, (5pi)/4)`

Hence proved.