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]}