we have the function
[tex]f(x)=\frac{1}{x^2+1}[/tex]
Find out the first derivative
[tex]f^{\prime}(x)=-\frac{2x}{(x^2+1)^2}[/tex]
Equate the first derivative to zero to calculate the critical points
[tex]\begin{gathered} -\frac{2x}{(x^2+1)^2}=0 \\ \\ -2x=0 \\ x=0 \end{gathered}[/tex]
The critical point is at x=0
Evaluate the first derivative at the intervals
(-infinite, 0) ----------> f'(x) is positive -------> f(x) is increasing
(0, infinite) -------> f'(x) is negative --------> f(x) is decreasing
that means
x=0 is a maximum
Find out the y-coordinate of the maximum
For x=0
[tex]\begin{gathered} f(x)=\frac{1}{0^2+1} \\ \\ f(x)=1 \end{gathered}[/tex]
The maximum is the point (0,1)
