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

Problem

ƒ(x)=x+1/x

Solution

1. Identify the domain of the function ƒ(x)=x+1/x The function is undefined at x=0 so the domain is (−∞,0)∪(0,∞)

2. Find the derivative of the function with respect to x using the power rule. Rewrite the function as ƒ(x)=x+x(−1)

ƒ(x)′=(d(x)+x(−1))/d(x)

ƒ(x)′=1−x(−2)

ƒ(x)′=1−1/(x2)

3. Set the derivative equal to zero to find potential critical points where the slope of the tangent line is horizontal.

1−1/(x2)=0

4. Solve for x by isolating the variable.

1=1/(x2)

x2=1

x=±1

5. Check for points where the derivative is undefined. The derivative ƒ(x)′=1−1/(x2) is undefined at x=0 However, since x=0 is not in the domain of the original function, it is not a critical point.

Final Answer

x=−1,1

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