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

1 Answer
Dec 29, 2017

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

Explanation:

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