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

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Answer 1 · 2016-12-12T13:57:49

decreasing.

Start by finding the derivative.

#f'(x) = (cosx xx e^x - sinx xx e^x)/(e^x)^2#

#f'(x) = (e^x(cosx - sinx))/((e^x)(e^x))#

#f'(x) = (cosx- sinx)/e^x#

The function will be increasing at #x = a# if #f'(a) > 0# and decreasing if #f'(a) < 0#.

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

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

This will be negative, so the function is decreasing.

Hopefully this helps!