Is #f(x)=e^x+cosx-e^xsinx# increasing or decreasing at #x=pi/6#?

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Answer 1 · 2017-06-07T19:09:31

Locally decreasing function.

#f'(x) = e^x - sin x - e^x sin x - e^x cos x#

#f'(pi/6) = e^(pi/6)(1 - sin frac{pi}{6} - cos frac{pi}{6}) - sin frac{pi}{6} #

#f'(pi/6) = e^(pi/6)(1 - 1/2 - sqrt 3 / 2) - 1/2 < 0 #

#e^(pi/6)(1/2 - sqrt 3 / 2) < 1/2#

#e^(pi/6)(1 - sqrt 3) < 1#

#e^(pi/6)(-1) < 1/(sqrt 3 - 1)#

#e^(pi/6) > -1/(sqrt 3 - 1)#

positive > negative