Calculus Examples

Find the Absolute Max and Min over the Interval g(x)=(6x^2)/(x-2) , [-2,1]
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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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.1.3
Differentiate.
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Step 1.1.1.3.1
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.2
Move to the left of .
Step 1.1.1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.6
Combine fractions.
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Step 1.1.1.3.6.1
Add and .
Step 1.1.1.3.6.2
Multiply by .
Step 1.1.1.3.6.3
Combine and .
Step 1.1.1.4
Simplify.
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Step 1.1.1.4.1
Apply the distributive property.
Step 1.1.1.4.2
Apply the distributive property.
Step 1.1.1.4.3
Apply the distributive property.
Step 1.1.1.4.4
Simplify the numerator.
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Step 1.1.1.4.4.1
Simplify each term.
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Step 1.1.1.4.4.1.1
Multiply by by adding the exponents.
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Step 1.1.1.4.4.1.1.1
Move .
Step 1.1.1.4.4.1.1.2
Multiply by .
Step 1.1.1.4.4.1.2
Multiply by .
Step 1.1.1.4.4.1.3
Multiply by .
Step 1.1.1.4.4.1.4
Multiply by .
Step 1.1.1.4.4.1.5
Multiply by .
Step 1.1.1.4.4.2
Subtract from .
Step 1.1.1.4.5
Factor out of .
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Step 1.1.1.4.5.1
Factor out of .
Step 1.1.1.4.5.2
Factor out of .
Step 1.1.1.4.5.3
Factor out of .
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
Set the numerator equal to zero.
Step 1.2.3
Solve the equation for .
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Step 1.2.3.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.3.2
Set equal to .
Step 1.2.3.3
Set equal to and solve for .
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Step 1.2.3.3.1
Set equal to .
Step 1.2.3.3.2
Add to both sides of the equation.
Step 1.2.3.4
The final solution is all the values that make true.
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
Set the equal to .
Step 1.3.2.2
Add to both sides of the equation.
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
Cancel the common factor of and .
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Step 1.4.1.2.1.1
Factor out of .
Step 1.4.1.2.1.2
Cancel the common factors.
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Step 1.4.1.2.1.2.1
Factor out of .
Step 1.4.1.2.1.2.2
Factor out of .
Step 1.4.1.2.1.2.3
Factor out of .
Step 1.4.1.2.1.2.4
Cancel the common factor.
Step 1.4.1.2.1.2.5
Rewrite the expression.
Step 1.4.1.2.2
Simplify the expression.
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Step 1.4.1.2.2.1
Raising to any positive power yields .
Step 1.4.1.2.2.2
Subtract from .
Step 1.4.1.2.2.3
Multiply by .
Step 1.4.1.2.2.4
Divide by .
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
Cancel the common factor of and .
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Step 1.4.2.2.1.1
Factor out of .
Step 1.4.2.2.1.2
Cancel the common factors.
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Step 1.4.2.2.1.2.1
Factor out of .
Step 1.4.2.2.1.2.2
Factor out of .
Step 1.4.2.2.1.2.3
Factor out of .
Step 1.4.2.2.1.2.4
Cancel the common factor.
Step 1.4.2.2.1.2.5
Rewrite the expression.
Step 1.4.2.2.2
Simplify the expression.
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Step 1.4.2.2.2.1
Raise to the power of .
Step 1.4.2.2.2.2
Subtract from .
Step 1.4.2.2.2.3
Multiply by .
Step 1.4.2.2.2.4
Divide by .
Step 1.4.3
Evaluate at .
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Step 1.4.3.1
Substitute for .
Step 1.4.3.2
Simplify.
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Step 1.4.3.2.1
Subtract from .
Step 1.4.3.2.2
The expression contains a division by . The expression is undefined.
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
Cancel the common factor of and .
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Step 3.1.2.1.1
Factor out of .
Step 3.1.2.1.2
Cancel the common factors.
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Step 3.1.2.1.2.1
Factor out of .
Step 3.1.2.1.2.2
Factor out of .
Step 3.1.2.1.2.3
Factor out of .
Step 3.1.2.1.2.4
Cancel the common factor.
Step 3.1.2.1.2.5
Rewrite the expression.
Step 3.1.2.2
Simplify the expression.
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Step 3.1.2.2.1
Raise to the power of .
Step 3.1.2.2.2
Subtract from .
Step 3.1.2.2.3
Multiply by .
Step 3.1.2.2.4
Divide by .
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
One to any power is one.
Step 3.2.2.2
Subtract from .
Step 3.2.2.3
Multiply by .
Step 3.2.2.4
Divide by .
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