Show that f(x) = cos2 x is a decreasing function on (0, π/2).

1.	Show that f(x) = cos2 x is a decreasing function on (0, π/2).
Answer» Given as f(x) = cos² x f'(x) = (d/dx)(cos² x) ⇒ f’(x) = 2 cos x (–sin x) ⇒ f’(x) = –2 sin (x) cos (x) ⇒ f’(x) = –sin2x As given x belongs to (0, π/2). ⇒ 2x ∈ (0, π) ⇒ Sin (2x)> 0 ⇒ –Sin (2x) < 0 ⇒ f’(x) < 0 Thus, condition for f(x) to be decreasing Hence f(x) is decreasing on interval (0, π/2). Thus proved

Answer»

Given as f(x) = cos² x

f'(x) = (d/dx)(cos² x)

⇒ f’(x) = 2 cos x (–sin x)

⇒ f’(x) = –2 sin (x) cos (x)

⇒ f’(x) = –sin2x

As given x belongs to (0, π/2).

⇒ 2x ∈ (0, π)

⇒ Sin (2x)> 0

⇒ –Sin (2x) < 0

⇒ f’(x) < 0

Thus, condition for f(x) to be decreasing

Hence f(x) is decreasing on interval (0, π/2).

Thus proved

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