Question

Find critical points and extrema points by using 1st derivative test. F(x) =1/x^2 + 1

Answer 1

we have the function

`f(x)=(1)/(x^2+1)`

Find out the first derivative

`f^(\prime)(x)=-(2x)/((x^2+1)^2)`

Equate the first derivative to zero to calculate the critical points

`\begin{gathered} -(2x)/((x^2+1)^2)=0 \\ \\ -2x=0 \\ x=0 \end{gathered}`

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

`\begin{gathered} f(x)=(1)/(0^2+1) \\ \\ f(x)=1 \end{gathered}`

The maximum is the point (0,1)