# Find the Absolute Max and Min over the Interval f(x)=x+1/x , [0.2,4]

,
Find the first derivative.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Rewrite the expression using the negative exponent rule .
Reorder terms.
Set the first derivative equal to zero.
Solve to find the critical points.
Subtract from both sides of the equation.
Solve for .
Multiply each term by and simplify.
Multiply each term in by .
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Rewrite as .
Rewrite the equation as .
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Any root of is .
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.
Evaluate the function at .
Simplify the right side.
Divide by .
Evaluate the function at .
Simplify the right side.
Divide by .
Evaluate the function at .
Simplify the right side.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .