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

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

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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)

User Yery
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