# How do you find the maximum, minimum and inflection points and concavity for the function #y = xe^(-x)#?

##### 1 Answer

There is a local maximum at

There is a non-stationary point of inflection at

#### Explanation:

We have:

# y = xe^(-x) #

graph{xe^(-x) [-5, 10, -5, 5]}

Firstly, let us look for critical points, that is coordinates where

# dy/dx = (x)(-e^(-x)) + (1)(e^(-x)) #

# " " = e^(-x) - xe^(-x) #

# " " = (1-x)e^(-x) #

For a critical point:

# dy/dx = 0 => (1-x)e^(-x) = 0 #

Which has a single solution

#x=1 => y = e^(-1) #

Thus there is a single critical point at

# (d^2y)/(dx^2) = -e^(-x) - {(1-x)e^(-x)} #

# " " = -(2-x)e^(-x) #

# " " = (x-2)e^(-x) #

When

As the second derivative is negative at the critical point then we can conclude the critical point

Secondly, we look for inflection points, which are coordinates where the second derivative vanishes:

# (d^2y)/(dx^2) = 0 => (x-2)e^(-x) = 0 #

Which has a single solution

# y = 2e^(-2) #

We already know that **not** correspond to a critical point and thus we have a non-stationary point of inflection at