Answer :
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 2nd and 4th 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
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