Find the Critical Points f(x)=1/(x^2+1)

Problem

ƒ(x)=1/(x^2+1)

Solution

1. Identify the domain of the function. Since x^2+1 is never zero for any real number x the domain of ƒ(x) is all real numbers, (−∞,∞)

2. Differentiate the function using the power rule or quotient rule. Rewrite the function as ƒ(x)=(x^2+1)^(−1) to apply the chain rule.

ƒ(x)^′=−1*(x^2+1)^(−2)⋅d(x^2+1)/d(x)

3. Simplify the derivative expression.

ƒ(x)^′=(−2*x)/((x^2+1)^2)

4. Set the derivative equal to zero to find the critical points. A fraction is zero only when its numerator is zero.

−2*x=0

5. Solve for x

x=0

6. Check for points where the derivative is undefined. Since the denominator (x^2+1)^2 is always at least 1 the derivative is defined for all real numbers.

Final Answer

x=0

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