# Is #f(x)=(x^2e^x)/(x+2)# increasing or decreasing at #x=-1#?

##### 1 Answer

Thus,

#### Explanation:

To determine if

Here, you can use the quotient rule:

if

#f'(x) = (g'(x) h(x) - g(x) h'(x)) /(h^2(x))#

For you,

The derivatives of

#g'(x) = x^2e^x + 2xe^x# (with the product rule)

#h'(x) = 1#

Thus, your derivative is:

#f'(x) = ((x^2e^x + 2xe^x)(x+2) - x^2e^x * 1) / (x+2)^2#

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

# = (e^x[(x^2 + 2x)(x+2) - x^2]) / (x+2)^2#

# = (e^x(x^3 - 3x^2 + 4x)) / (x+2)^2#

Now, let's evaluate the derivative at

#f'(-1) = (e^(-1)((-1)^3 - 3(-1)^2 +4*(-1))) / (-1+2)^2#

# = e^(-1)(-1 - 3 -4) #

# = -8 e^(-1)#

As

#f'(-1) = -8 e^(-1) < 0# .

Thus,