Find the intervals in which the function f is given by f(x) = sin x – cos x, 0 ≤ x ≤ 2π

is strictly increasing or strictly decreasing.

Know that, a function f(x) is said to be strictly increasing function on (a, b) if x₁ < x₂⇒ f(x₁) < f(x₂) for all x₁, x₂∈ (a, b).

If x₁ < x₂⇒ f(x₁) > f(x₂) for all x₁, x₂∈ (a, b), then f(x) is strictly decreasing on (a, b).

We have,

f(x) = sin x – cos x, 0 ≤ x ≤ 2π

Differentiating f(x) with respect to x,

…(i)

[∵, if y = sin x ⇒ y’ = cos x and if y = cos x ⇒ y’ = -sin x]

For critical points,

⇒ cos x + sin x = 0

⇒ sin x = -cos x

⇒ tan x = -1

But, 0 ≤ x ≤ 2π

So,

We can’t take .

We should know that, the necessary sufficient condition for a differential function defined on (a, b) to be strictly increasing on (a, b) is that f’(x) > 0 for all x ∈ (a, b).

For ,

Taking sine on both sides and then multiplying by √2,

⇒ 1 < f’(x) < 0

∴, the function is decreasing at

For ,

Taking sine on both sides and then multiplying by √2,

⇒ 0 < f’(x) < 0

∴, the function is decreasing at .

For ,

Taking sine on both sides and then multiplying by √2,

⇒ 0 < f’(x) < 1

∴, the function is increasing at

We can conclude that,

The function is strictly increasing at .

The function is strictly decreasing at .