# How do you sketch the graph #y=1/(1+x^4)# using the first and second derivatives?

##### 1 Answer

#### Explanation:

To analyze the behavior of the function we start from considering the domain of definition and see that

We can also see that

so that its graph is symmetrical with respect to the

At the limits of the domain

so the function is going to have

Now we calculate the first and second derivatives:

We can see that the only critical point where

# x<0 => y'(x) >0#

# x>0 => y'(x) <0#

so that the function is monotone increasing for

The second derivative has three zeros in

#abs(x) > root(4)(3/5) => y''(x) > 0# and y(x) is concave up.

#abs(x) < root(4)(3/5) => y''(x) < 0# and y(x) is concave down.

graph{1/(1+x^4) [-10, 10, -5, 5]}