In order to find if a function is increasing or decreasing, we take its derivative and evaluate it at the #x#-value in question. If the derivative is positive at that point, the function is increasing; if the derivative is negative, the function is decreasing.
Step 1: Find the Derivative
Since we have #x-2# divided by #e^x#, we need to use the quotient rule to take the derivative, which states:
#d/dx(u/v) = (u'v-uv')/v^2#
Where #u# and #v# are functions of #x#.
In our case, we have #u=x-2# and #v=e^x#. Taking the derivative of these two:
#u' = 1#
#v' = e^x#
Now we can substitute into the quotient rule:
#d/dx((x-2)/e^x) = ((x-2)'(e^x)-(x-2)(e^x)')/(e^x)^2#
#= (1(e^x)-(x-2)(e^x))/(e^(2x)#
We can do a little simplifying here:
#= (e^x(1-(x-2)))/(e^(2x)#
#= (1-x+2)/(e^x)#
#= (-x+3)/(e^x)#
Step 2: Evaluate
We are being asked to find if the function is increasing or decreasing at #x=-2#; that means we evaluate the derivative at #x=-2#:
#=(-(-2)+3)/(e^(-2))#
#=(5)/(e^(-2))=5e^2#
We don't even need to find #5e^2#, because it is definitely positive. And because it's positive, we can say that the function #(x-2)/e^x# is increasing at #x=-2#. To confirm, take a look at the graph of #(x-2)/e^x# and you will see it increasing at #x=-2#.
graph{(x-2)/e^x [-10, 10, -5, 5]}
