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Question
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = sinx − cos x, 0 < x < 2π
Solution
Given function f(x) = sin x - cos x, 0 < x < 2`pi`
∴ f'(x) = cos x + sin x = cos x (1 + tan x)
If f'(x) = 0 then 1 + tan x = 0
= tan x = - 1
= x = `(3pi)/4, (7pi)/4`
Now f''(x) = `d/dx (cos x + sin x) = - sin x + cos x`
at `x = (3 pi)/4 f' (x) = -sin (3pi)/4 + cos (3pi)/4`
`= - (1/sqrt4) - 1/sqrt2`
`= - 2/sqrt2`
`= - sqrt2` ...(negative)
∴ f(x) is maximum at `x = (3pi)/4`.
and the maximum value of f(x)
`f((3pi)/4)= sin (3pi)/4 - cos (3pi)/4`
`= 1/sqrt2 - (- 1/sqrt2)`
`= 2/sqrt2`
`= sqrt2`
Again, at `x = (7pi)/4 f' (x) = -sin (7pi)/4 + cos (7pi)/4`
`= - ((-1)/sqrt2) + 1/sqrt2`
`= 2/sqrt2`
`= sqrt2` ... (positive)
∴ f(x) is minimum at `x = (7 pi)/4`.
and the minimum value of f(x)
`= f ((7pi)/4) = sin ((7pi)/4) - cos ((7pi)/4)`
`= - 1/sqrt2 - 1/sqrt2`
`= - sqrt2`
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