Is #f(x)=x-e^x/sinx# increasing or decreasing at #x=pi/3#?

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Answer 1 · 2017-12-29T17:20:38

#f'(pi/3)<0=>quad# f is decreasing

#f(x)=x−e^x/sinx#

#f'(x)=1−(e^x*sinx-e^x*cosx)/sin^2x#

#f'(x)=1−(e^x(sinx-cosx))/sin^2x#

#e^x>0#

#f'(pi/3)=1−(e^(pi/3)(sin(pi/3)-cos(pi/3)))/sin^2(pi/3)#

#=>1-(e^(pi/3)(sqrt2/2-1/2))/(sqrt2/2)^2#

#=>1-e^(pi/3)(sqrt2-1)#

Aproxximately:

#(sqrt2-1)~~0.414#

#pi/3~~3.15/3~~1.05#

#e^1.05~~e^(1+5/100)>2.71#

#2.8xx0.414~~2.8xx2/5>1#

#=>1-1.1#

#f'(pi/3)<0=>quad# f is decreasing