Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^3-5x-1 ; between 2 and 4
; between and
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
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Multiply by .
Step 1.1.1.3
Differentiate using the Constant Rule.
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Step 1.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.2
Add and .
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
Add to both sides of the equation.
Step 1.2.3
Divide each term in by and simplify.
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Step 1.2.3.1
Divide each term in by .
Step 1.2.3.2
Simplify the left side.
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Step 1.2.3.2.1
Cancel the common factor of .
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Step 1.2.3.2.1.1
Cancel the common factor.
Step 1.2.3.2.1.2
Divide by .
Step 1.2.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.5
Simplify .
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Step 1.2.5.1
Rewrite as .
Step 1.2.5.2
Multiply by .
Step 1.2.5.3
Combine and simplify the denominator.
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Step 1.2.5.3.1
Multiply by .
Step 1.2.5.3.2
Raise to the power of .
Step 1.2.5.3.3
Raise to the power of .
Step 1.2.5.3.4
Use the power rule to combine exponents.
Step 1.2.5.3.5
Add and .
Step 1.2.5.3.6
Rewrite as .
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Step 1.2.5.3.6.1
Use to rewrite as .
Step 1.2.5.3.6.2
Apply the power rule and multiply exponents, .
Step 1.2.5.3.6.3
Combine and .
Step 1.2.5.3.6.4
Cancel the common factor of .
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Step 1.2.5.3.6.4.1
Cancel the common factor.
Step 1.2.5.3.6.4.2
Rewrite the expression.
Step 1.2.5.3.6.5
Evaluate the exponent.
Step 1.2.5.4
Simplify the numerator.
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Step 1.2.5.4.1
Combine using the product rule for radicals.
Step 1.2.5.4.2
Multiply by .
Step 1.2.6
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.6.1
First, use the positive value of the to find the first solution.
Step 1.2.6.2
Next, use the negative value of the to find the second solution.
Step 1.2.6.3
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
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
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
Simplify the numerator.
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Step 1.4.1.2.1.2.1
Rewrite as .
Step 1.4.1.2.1.2.2
Raise to the power of .
Step 1.4.1.2.1.2.3
Rewrite as .
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Step 1.4.1.2.1.2.3.1
Factor out of .
Step 1.4.1.2.1.2.3.2
Rewrite as .
Step 1.4.1.2.1.2.4
Pull terms out from under the radical.
Step 1.4.1.2.1.3
Raise to the power of .
Step 1.4.1.2.1.4
Cancel the common factor of and .
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Step 1.4.1.2.1.4.1
Factor out of .
Step 1.4.1.2.1.4.2
Cancel the common factors.
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Step 1.4.1.2.1.4.2.1
Factor out of .
Step 1.4.1.2.1.4.2.2
Cancel the common factor.
Step 1.4.1.2.1.4.2.3
Rewrite the expression.
Step 1.4.1.2.1.5
Combine and .
Step 1.4.1.2.1.6
Move the negative in front of the fraction.
Step 1.4.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.4.1.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.4.1.2.3.1
Multiply by .
Step 1.4.1.2.3.2
Multiply by .
Step 1.4.1.2.4
Combine the numerators over the common denominator.
Step 1.4.1.2.5
Simplify each term.
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Step 1.4.1.2.5.1
Simplify the numerator.
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Step 1.4.1.2.5.1.1
Multiply by .
Step 1.4.1.2.5.1.2
Subtract from .
Step 1.4.1.2.5.2
Move the negative in front of the fraction.
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
Simplify each term.
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Step 1.4.2.2.1.1
Use the power rule to distribute the exponent.
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Step 1.4.2.2.1.1.1
Apply the product rule to .
Step 1.4.2.2.1.1.2
Apply the product rule to .
Step 1.4.2.2.1.2
Raise to the power of .
Step 1.4.2.2.1.3
Simplify the numerator.
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Step 1.4.2.2.1.3.1
Rewrite as .
Step 1.4.2.2.1.3.2
Raise to the power of .
Step 1.4.2.2.1.3.3
Rewrite as .
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Step 1.4.2.2.1.3.3.1
Factor out of .
Step 1.4.2.2.1.3.3.2
Rewrite as .
Step 1.4.2.2.1.3.4
Pull terms out from under the radical.
Step 1.4.2.2.1.4
Raise to the power of .
Step 1.4.2.2.1.5
Cancel the common factor of and .
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Step 1.4.2.2.1.5.1
Factor out of .
Step 1.4.2.2.1.5.2
Cancel the common factors.
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Step 1.4.2.2.1.5.2.1
Factor out of .
Step 1.4.2.2.1.5.2.2
Cancel the common factor.
Step 1.4.2.2.1.5.2.3
Rewrite the expression.
Step 1.4.2.2.1.6
Multiply .
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Step 1.4.2.2.1.6.1
Multiply by .
Step 1.4.2.2.1.6.2
Combine and .
Step 1.4.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.4.2.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.4.2.2.3.1
Multiply by .
Step 1.4.2.2.3.2
Multiply by .
Step 1.4.2.2.4
Combine the numerators over the common denominator.
Step 1.4.2.2.5
Simplify the numerator.
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Step 1.4.2.2.5.1
Multiply by .
Step 1.4.2.2.5.2
Add and .
Step 1.4.3
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
Simplify each term.
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Step 3.1.2.1.1
Raise to the power of .
Step 3.1.2.1.2
Multiply by .
Step 3.1.2.2
Simplify by subtracting numbers.
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Step 3.1.2.2.1
Subtract from .
Step 3.1.2.2.2
Subtract from .
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
Simplify each term.
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Step 3.2.2.1.1
Raise to the power of .
Step 3.2.2.1.2
Multiply by .
Step 3.2.2.2
Simplify by subtracting numbers.
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Step 3.2.2.2.1
Subtract from .
Step 3.2.2.2.2
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