Is f(x)=cosx/e^xf(x)=cosxex increasing or decreasing at x=pi/6x=π6?

1 Answer
Dec 2, 2015

f(x)f(x) is decreasing at x=pi/6x=π6.

Explanation:

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)f(x), then evaluate it at x=pi/6x=π6, 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.