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

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Answer 1 · 2015-12-02T08:48:35

#f(x)# is decreasing at #x=pi/6#.

A function is increasing or decreasing at a given point if its first derivative, evaluated at that point, is positive or negative respectively.

To answer, then, we first must find the derivative of #f(x)#, then evaluate it at #x=pi/6#, and finally check the sign of the result.

First, let's use the quotient rule to evaluate the derivative:
#f'(x) = d/dxcos(x)/e^x#

# = (-sin(x)e^x - cos(x)e^x)/(e^x)^2#

#=(-e^x(cos(x)+sin(x)))/(e^x)^2#

#= -(cos(x)+sin(x))/e^x#

Next, we evaluate the first derivative at #x=pi/6#

#f'(pi/6) = -(cos(pi/6)+sin(pi/6))/e^(pi/6)#

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

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

Finally, we check the sign of the result.

#-sqrt(3) - 1 < 0# and #2e^(pi/6) > 0#

thus, as the quotient of a positive number and a negative number is negative,

#f'(pi/6) < 0#

meaning #f(x)# is decreasing at #x=pi/6#.