Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^2+320/x ; (0,infinity)
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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
Simplify.
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Step 1.1.1.3.1
Rewrite the expression using the negative exponent rule .
Step 1.1.1.3.2
Combine terms.
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Step 1.1.1.3.2.1
Combine and .
Step 1.1.1.3.2.2
Move the negative in front of the fraction.
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
Find the LCD of the terms in the equation.
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Step 1.2.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2.2.2
The LCM of one and any expression is the expression.
Step 1.2.3
Multiply each term in by to eliminate the fractions.
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Step 1.2.3.1
Multiply each term in by .
Step 1.2.3.2
Simplify the left side.
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Step 1.2.3.2.1
Simplify each term.
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Step 1.2.3.2.1.1
Multiply by by adding the exponents.
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Step 1.2.3.2.1.1.1
Move .
Step 1.2.3.2.1.1.2
Multiply by .
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Step 1.2.3.2.1.1.2.1
Raise to the power of .
Step 1.2.3.2.1.1.2.2
Use the power rule to combine exponents.
Step 1.2.3.2.1.1.3
Add and .
Step 1.2.3.2.1.2
Cancel the common factor of .
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Step 1.2.3.2.1.2.1
Move the leading negative in into the numerator.
Step 1.2.3.2.1.2.2
Cancel the common factor.
Step 1.2.3.2.1.2.3
Rewrite the expression.
Step 1.2.3.3
Simplify the right side.
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Step 1.2.3.3.1
Multiply by .
Step 1.2.4
Solve the equation.
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Step 1.2.4.1
Add to both sides of the equation.
Step 1.2.4.2
Subtract from both sides of the equation.
Step 1.2.4.3
Factor out of .
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Step 1.2.4.3.1
Factor out of .
Step 1.2.4.3.2
Factor out of .
Step 1.2.4.3.3
Factor out of .
Step 1.2.4.4
Divide each term in by and simplify.
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Step 1.2.4.4.1
Divide each term in by .
Step 1.2.4.4.2
Simplify the left side.
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Step 1.2.4.4.2.1
Cancel the common factor of .
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Step 1.2.4.4.2.1.1
Cancel the common factor.
Step 1.2.4.4.2.1.2
Divide by .
Step 1.2.4.4.3
Simplify the right side.
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Step 1.2.4.4.3.1
Divide by .
Step 1.2.4.5
Add to both sides of the equation.
Step 1.2.4.6
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.4.7
Simplify .
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Step 1.2.4.7.1
Rewrite as .
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Step 1.2.4.7.1.1
Factor out of .
Step 1.2.4.7.1.2
Rewrite as .
Step 1.2.4.7.2
Pull terms out from under the radical.
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
Simplify each term.
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Step 1.4.1.2.1.1
Apply the product rule to .
Step 1.4.1.2.1.2
Raise to the power of .
Step 1.4.1.2.1.3
Rewrite as .
Step 1.4.1.2.1.4
Raise to the power of .
Step 1.4.1.2.1.5
Rewrite as .
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Step 1.4.1.2.1.5.1
Factor out of .
Step 1.4.1.2.1.5.2
Rewrite as .
Step 1.4.1.2.1.6
Pull terms out from under the radical.
Step 1.4.1.2.1.7
Multiply by .
Step 1.4.1.2.1.8
Cancel the common factor of and .
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Step 1.4.1.2.1.8.1
Factor out of .
Step 1.4.1.2.1.8.2
Cancel the common factors.
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Step 1.4.1.2.1.8.2.1
Factor out of .
Step 1.4.1.2.1.8.2.2
Cancel the common factor.
Step 1.4.1.2.1.8.2.3
Rewrite the expression.
Step 1.4.1.2.1.9
Multiply by .
Step 1.4.1.2.1.10
Combine and simplify the denominator.
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Step 1.4.1.2.1.10.1
Multiply by .
Step 1.4.1.2.1.10.2
Raise to the power of .
Step 1.4.1.2.1.10.3
Use the power rule to combine exponents.
Step 1.4.1.2.1.10.4
Add and .
Step 1.4.1.2.1.10.5
Rewrite as .
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Step 1.4.1.2.1.10.5.1
Use to rewrite as .
Step 1.4.1.2.1.10.5.2
Apply the power rule and multiply exponents, .
Step 1.4.1.2.1.10.5.3
Combine and .
Step 1.4.1.2.1.10.5.4
Cancel the common factor of .
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Step 1.4.1.2.1.10.5.4.1
Cancel the common factor.
Step 1.4.1.2.1.10.5.4.2
Rewrite the expression.
Step 1.4.1.2.1.10.5.5
Evaluate the exponent.
Step 1.4.1.2.1.11
Cancel the common factor of and .
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Step 1.4.1.2.1.11.1
Factor out of .
Step 1.4.1.2.1.11.2
Cancel the common factors.
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Step 1.4.1.2.1.11.2.1
Factor out of .
Step 1.4.1.2.1.11.2.2
Cancel the common factor.
Step 1.4.1.2.1.11.2.3
Rewrite the expression.
Step 1.4.1.2.1.11.2.4
Divide by .
Step 1.4.1.2.1.12
Rewrite as .
Step 1.4.1.2.1.13
Raise to the power of .
Step 1.4.1.2.1.14
Rewrite as .
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Step 1.4.1.2.1.14.1
Factor out of .
Step 1.4.1.2.1.14.2
Rewrite as .
Step 1.4.1.2.1.15
Pull terms out from under the radical.
Step 1.4.1.2.1.16
Multiply 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
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 1.4.3
List all of the points.
Step 2
Use the first derivative test to determine which points can be maxima or minima.
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Step 2.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 2.2
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 2.2.1
Replace the variable with in the expression.
Step 2.2.2
Simplify the result.
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Step 2.2.2.1
Simplify each term.
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Step 2.2.2.1.1
Multiply by .
Step 2.2.2.1.2
Raise to the power of .
Step 2.2.2.1.3
Divide by .
Step 2.2.2.1.4
Multiply by .
Step 2.2.2.2
Subtract from .
Step 2.2.2.3
The final answer is .
Step 2.3
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 2.3.1
Replace the variable with in the expression.
Step 2.3.2
Simplify the result.
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Step 2.3.2.1
Simplify each term.
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Step 2.3.2.1.1
Multiply by .
Step 2.3.2.1.2
Raise to the power of .
Step 2.3.2.1.3
Divide by .
Step 2.3.2.1.4
Multiply by .
Step 2.3.2.2
Subtract from .
Step 2.3.2.3
The final answer is .
Step 2.4
Since the first derivative changed signs from negative to positive around , then is a local minimum.
is a local minimum
is a local minimum
Step 3
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.
No absolute maximum
Absolute Minimum:
Step 4