Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x+16/x , [-7,-1]
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Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
Differentiate.
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Step 1.1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2
Evaluate .
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Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Rewrite as .
Step 1.1.1.2.3
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.4
Multiply by .
Step 1.1.1.3
Rewrite the expression using the negative exponent rule .
Step 1.1.1.4
Simplify.
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Step 1.1.1.4.1
Combine terms.
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Step 1.1.1.4.1.1
Combine and .
Step 1.1.1.4.1.2
Move the negative in front of the fraction.
Step 1.1.1.4.2
Reorder terms.
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Subtract from both sides of the equation.
Step 1.2.3
Find the LCD of the terms in the equation.
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Step 1.2.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2.3.2
The LCM of one and any expression is the expression.
Step 1.2.4
Multiply each term in by to eliminate the fractions.
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Step 1.2.4.1
Multiply each term in by .
Step 1.2.4.2
Simplify the left side.
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Step 1.2.4.2.1
Cancel the common factor of .
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Step 1.2.4.2.1.1
Move the leading negative in into the numerator.
Step 1.2.4.2.1.2
Cancel the common factor.
Step 1.2.4.2.1.3
Rewrite the expression.
Step 1.2.5
Solve the equation.
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Step 1.2.5.1
Rewrite the equation as .
Step 1.2.5.2
Divide each term in by and simplify.
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Step 1.2.5.2.1
Divide each term in by .
Step 1.2.5.2.2
Simplify the left side.
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Step 1.2.5.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.5.2.2.2
Divide by .
Step 1.2.5.2.3
Simplify the right side.
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Step 1.2.5.2.3.1
Divide by .
Step 1.2.5.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.5.4
Simplify .
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Step 1.2.5.4.1
Rewrite as .
Step 1.2.5.4.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.5.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.5.5.1
First, use the positive value of the to find the first solution.
Step 1.2.5.5.2
Next, use the negative value of the to find the second solution.
Step 1.2.5.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.3
Find the values where the derivative is undefined.
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Step 1.3.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.3.2
Solve for .
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Step 1.3.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.3.2.2
Simplify .
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Step 1.3.2.2.1
Rewrite as .
Step 1.3.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.3.2.2.3
Plus or minus is .
Step 1.4
Evaluate at each value where the derivative is or undefined.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
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Step 1.4.1.2.1
Divide by .
Step 1.4.1.2.2
Add and .
Step 1.4.2
Evaluate at .
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Step 1.4.2.1
Substitute for .
Step 1.4.2.2
Simplify.
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Step 1.4.2.2.1
Divide by .
Step 1.4.2.2.2
Subtract from .
Step 1.4.3
Evaluate at .
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Step 1.4.3.1
Substitute for .
Step 1.4.3.2
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 1.4.4
List all of the points.
Step 2
Exclude the points that are not on the interval.
Step 3
Evaluate at the included endpoints.
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Step 3.1
Evaluate at .
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Step 3.1.1
Substitute for .
Step 3.1.2
Simplify.
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Step 3.1.2.1
Move the negative in front of the fraction.
Step 3.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 3.1.2.3
Combine and .
Step 3.1.2.4
Combine the numerators over the common denominator.
Step 3.1.2.5
Simplify the numerator.
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Step 3.1.2.5.1
Multiply by .
Step 3.1.2.5.2
Subtract from .
Step 3.1.2.6
Move the negative in front of the fraction.
Step 3.2
Evaluate at .
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Step 3.2.1
Substitute for .
Step 3.2.2
Simplify.
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Step 3.2.2.1
Divide by .
Step 3.2.2.2
Subtract from .
Step 3.3
List all of the points.
Step 4
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
Absolute Minimum:
Step 5