f(x) = cos x - sin x
Differentiating with respect to x
f'(x) = - sin x - cos x
f''(x) = - cos x + sin x
For extreme values of f(x) ,
f'(x) = 0
=> - sin x - cos x = 0
=> sin x = - cos x
=> tan x = -1
=> tan x = tan (−4π)
=> x = nπ −4π Where n is an integer
Since x ∈(−2π,2π) the only value of n is 0.
So x = −4π
f''(−4π)=−cos(−4π)+sin(−4π)
= −cos(4π)−sin(4π)
= −21−21
= −22
= −2
So f''(−4π) is negative.
Therefore f(x) has a maximum value at x = −4π and maximum value is f(−4π)=cos(−4π)−sin(−4π)
= 21+21
=2
Answer:
Yes , there is a extreme value of f(x) = cos x - sin x at x = −4π and it is a maximum value equating 2
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