find the local maximA or minima of the function: f(x)= sinx + cosx, where 0
Answers
Given : f(x) = Sin(x) + Cos(x)
To find : Local maxima and local minima
Solution:
f(x) = Sin(x) + Cos(x)
f'(x) = Cos(x) - Sin(x)
put f'(x) = 0
=> Cos(x) = Sin(x)
=> Tan (x) = 1
=> x = π/4 , 5π/4
f'(x) = Cos(x) - Sin(x)
f''(x) = -Sin(x) - Cos(x)
f''(x) = - ( Sin(x) + Cos(x) )
x = π/4 => f''(x) < 0
Hence local maxima at x = π/4
x = 5π/4 f''(x) > 0
Hence local minima at x = 5π/4
f(x) = Sin(x) + Cos(x)
f(π/4) = Sin(π/4) + Cos(π/4) = 1/√2 + 1/√2 = √2
f(5π/4) = Sin(5π/4) + Cos(5π/4) = -1/√2 - 1/√2 = -√2
Local maxima = √2
Local Minima = -√2
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