Find the local maxima and local minima, of the function f(x) sin x – cos x, 0< x < 2π .Also, find the local maximum and local minimum values.

Given,

f(x) = sin x – cos x

We know that for local maxima-

f’(x) = 0 and f’’(x) < 0

and for local minima-

f’(x) = 0 and f’’(x) > 0

∴ differentiating f(x) w.r.t x we get-

f’(x) = cos x – (-sin x)

⇒ f’(x) = sin x + cos x

For local maxima or minima-

f’(x) = 0

∴ sin x + cos x = 0

⇒ sin x = -cos x

⇒ tan x = -1 (it is negative in 2^nd and 4^th quadrant)

As 0<x<2π

∴ x =

To check for maxima and minima, we need to differentiate it again-

f’’(x) = cos x – sin x

putting x = 3π/4 –

⇒ f’’(3π/4) =

∴ x = 3π/4 is the point of maxima

putting x = 7π/4 –

⇒ f’’(7π/4) =

We can’t say any thing at this stage so we will differentiate it again and check the sign of next derivative.

∴ f’’’(x) = -sin x – cos x

And f’’’(7π/4) = √2 > 0

∴ x = 7π/4 is the point of minima.

∴ Local Maxima = f(3π/4) = sin (3π/4) – cos (3π/4) = √2

Local Minima = f(7π/4) = sin (7π/4) – cos (7π/4) = -√2