Calculus Examples

Find the Turning Points f(x)=-x^3-11x^2-28x
Step 1
Find the first derivative.
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Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
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Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
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Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Evaluate .
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Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.4.3
Multiply by .
Step 2
Set the first derivative equal to and solve for .
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Step 2.1
Use the quadratic formula to find the solutions.
Step 2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 2.3
Simplify.
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Step 2.3.1
Simplify the numerator.
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Step 2.3.1.1
Raise to the power of .
Step 2.3.1.2
Multiply .
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Step 2.3.1.2.1
Multiply by .
Step 2.3.1.2.2
Multiply by .
Step 2.3.1.3
Subtract from .
Step 2.3.1.4
Rewrite as .
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Step 2.3.1.4.1
Factor out of .
Step 2.3.1.4.2
Rewrite as .
Step 2.3.1.5
Pull terms out from under the radical.
Step 2.3.2
Multiply by .
Step 2.3.3
Simplify .
Step 2.3.4
Move the negative in front of the fraction.
Step 2.4
Simplify the expression to solve for the portion of the .
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Step 2.4.1
Simplify the numerator.
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Step 2.4.1.1
Raise to the power of .
Step 2.4.1.2
Multiply .
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Step 2.4.1.2.1
Multiply by .
Step 2.4.1.2.2
Multiply by .
Step 2.4.1.3
Subtract from .
Step 2.4.1.4
Rewrite as .
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Step 2.4.1.4.1
Factor out of .
Step 2.4.1.4.2
Rewrite as .
Step 2.4.1.5
Pull terms out from under the radical.
Step 2.4.2
Multiply by .
Step 2.4.3
Simplify .
Step 2.4.4
Move the negative in front of the fraction.
Step 2.4.5
Change the to .
Step 2.5
Simplify the expression to solve for the portion of the .
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Step 2.5.1
Simplify the numerator.
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Step 2.5.1.1
Raise to the power of .
Step 2.5.1.2
Multiply .
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Step 2.5.1.2.1
Multiply by .
Step 2.5.1.2.2
Multiply by .
Step 2.5.1.3
Subtract from .
Step 2.5.1.4
Rewrite as .
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Step 2.5.1.4.1
Factor out of .
Step 2.5.1.4.2
Rewrite as .
Step 2.5.1.5
Pull terms out from under the radical.
Step 2.5.2
Multiply by .
Step 2.5.3
Simplify .
Step 2.5.4
Move the negative in front of the fraction.
Step 2.5.5
Change the to .
Step 2.6
The final answer is the combination of both solutions.
Step 3
Split into separate intervals around the values that make the first derivative or undefined.
Step 4
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 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
Simplify each term.
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Step 4.2.1.1
Raise to the power of .
Step 4.2.1.2
Multiply by .
Step 4.2.1.3
Multiply by .
Step 4.2.2
Simplify by adding and subtracting.
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Step 4.2.2.1
Add and .
Step 4.2.2.2
Subtract from .
Step 4.2.3
The final answer is .
Step 5
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 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
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Step 5.2.1
Simplify each term.
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Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Multiply by .
Step 5.2.2
Simplify by adding and subtracting.
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Step 5.2.2.1
Add and .
Step 5.2.2.2
Subtract from .
Step 5.2.3
The final answer is .
Step 6
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 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
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Step 6.2.1
Simplify each term.
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Step 6.2.1.1
Raising to any positive power yields .
Step 6.2.1.2
Multiply by .
Step 6.2.1.3
Multiply by .
Step 6.2.2
Simplify by adding and subtracting.
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Step 6.2.2.1
Add and .
Step 6.2.2.2
Subtract from .
Step 6.2.3
The final answer is .
Step 7
Since the first derivative changed signs from negative to positive around , then there is a turning point at .
Step 8
Find the y-coordinate of to find the turning point.
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Step 8.1
Find to find the y-coordinate of .
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Step 8.1.1
Replace the variable with in the expression.
Step 8.1.2
Simplify .
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Step 8.1.2.1
Remove parentheses.
Step 8.1.2.2
Simplify each term.
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Step 8.1.2.2.1
Use the power rule to distribute the exponent.
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Step 8.1.2.2.1.1
Apply the product rule to .
Step 8.1.2.2.1.2
Apply the product rule to .
Step 8.1.2.2.2
Multiply by by adding the exponents.
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Step 8.1.2.2.2.1
Move .
Step 8.1.2.2.2.2
Multiply by .
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Step 8.1.2.2.2.2.1
Raise to the power of .
Step 8.1.2.2.2.2.2
Use the power rule to combine exponents.
Step 8.1.2.2.2.3
Add and .
Step 8.1.2.2.3
Raise to the power of .
Step 8.1.2.2.4
Multiply by .
Step 8.1.2.2.5
Raise to the power of .
Step 8.1.2.2.6
Use the Binomial Theorem.
Step 8.1.2.2.7
Simplify each term.
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Step 8.1.2.2.7.1
Raise to the power of .
Step 8.1.2.2.7.2
Raise to the power of .
Step 8.1.2.2.7.3
Multiply by .
Step 8.1.2.2.7.4
Multiply by .
Step 8.1.2.2.7.5
Rewrite as .
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Step 8.1.2.2.7.5.1
Use to rewrite as .
Step 8.1.2.2.7.5.2
Apply the power rule and multiply exponents, .
Step 8.1.2.2.7.5.3
Combine and .
Step 8.1.2.2.7.5.4
Cancel the common factor of .
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Step 8.1.2.2.7.5.4.1
Cancel the common factor.
Step 8.1.2.2.7.5.4.2
Rewrite the expression.
Step 8.1.2.2.7.5.5
Evaluate the exponent.
Step 8.1.2.2.7.6
Multiply by .
Step 8.1.2.2.7.7
Rewrite as .
Step 8.1.2.2.7.8
Raise to the power of .
Step 8.1.2.2.7.9
Rewrite as .
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Step 8.1.2.2.7.9.1
Factor out of .
Step 8.1.2.2.7.9.2
Rewrite as .
Step 8.1.2.2.7.10
Pull terms out from under the radical.
Step 8.1.2.2.8
Add and .
Step 8.1.2.2.9
Add and .
Step 8.1.2.2.10
Use the power rule to distribute the exponent.
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Step 8.1.2.2.10.1
Apply the product rule to .
Step 8.1.2.2.10.2
Apply the product rule to .
Step 8.1.2.2.11
Raise to the power of .
Step 8.1.2.2.12
Multiply by .
Step 8.1.2.2.13
Raise to the power of .
Step 8.1.2.2.14
Rewrite as .
Step 8.1.2.2.15
Expand using the FOIL Method.
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Step 8.1.2.2.15.1
Apply the distributive property.
Step 8.1.2.2.15.2
Apply the distributive property.
Step 8.1.2.2.15.3
Apply the distributive property.
Step 8.1.2.2.16
Simplify and combine like terms.
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Step 8.1.2.2.16.1
Simplify each term.
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Step 8.1.2.2.16.1.1
Multiply by .
Step 8.1.2.2.16.1.2
Move to the left of .
Step 8.1.2.2.16.1.3
Combine using the product rule for radicals.
Step 8.1.2.2.16.1.4
Multiply by .
Step 8.1.2.2.16.1.5
Rewrite as .
Step 8.1.2.2.16.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 8.1.2.2.16.2
Add and .
Step 8.1.2.2.16.3
Add and .
Step 8.1.2.2.17
Combine and .
Step 8.1.2.2.18
Move the negative in front of the fraction.
Step 8.1.2.2.19
Multiply .
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Step 8.1.2.2.19.1
Multiply by .
Step 8.1.2.2.19.2
Combine and .
Step 8.1.2.3
Find the common denominator.
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Step 8.1.2.3.1
Multiply by .
Step 8.1.2.3.2
Multiply by .
Step 8.1.2.3.3
Multiply by .
Step 8.1.2.3.4
Multiply by .
Step 8.1.2.3.5
Reorder the factors of .
Step 8.1.2.3.6
Multiply by .
Step 8.1.2.3.7
Multiply by .
Step 8.1.2.4
Combine the numerators over the common denominator.
Step 8.1.2.5
Simplify each term.
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Step 8.1.2.5.1
Apply the distributive property.
Step 8.1.2.5.2
Multiply by .
Step 8.1.2.5.3
Multiply by .
Step 8.1.2.5.4
Apply the distributive property.
Step 8.1.2.5.5
Multiply by .
Step 8.1.2.5.6
Multiply by .
Step 8.1.2.5.7
Apply the distributive property.
Step 8.1.2.5.8
Multiply by .
Step 8.1.2.5.9
Apply the distributive property.
Step 8.1.2.5.10
Multiply by .
Step 8.1.2.5.11
Multiply by .
Step 8.1.2.6
Simplify by adding terms.
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Step 8.1.2.6.1
Subtract from .
Step 8.1.2.6.2
Add and .
Step 8.1.2.6.3
Subtract from .
Step 8.1.2.6.4
Add and .
Step 8.2
Write the and coordinates in point form.
Step 9
Since the first derivative changed signs from positive to negative around , then there is a turning point at .
Step 10
Find the y-coordinate of to find the turning point.
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Step 10.1
Find to find the y-coordinate of .
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Step 10.1.1
Replace the variable with in the expression.
Step 10.1.2
Simplify .
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Step 10.1.2.1
Remove parentheses.
Step 10.1.2.2
Simplify each term.
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Step 10.1.2.2.1
Use the power rule to distribute the exponent.
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Step 10.1.2.2.1.1
Apply the product rule to .
Step 10.1.2.2.1.2
Apply the product rule to .
Step 10.1.2.2.2
Multiply by by adding the exponents.
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Step 10.1.2.2.2.1
Move .
Step 10.1.2.2.2.2
Multiply by .
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Step 10.1.2.2.2.2.1
Raise to the power of .
Step 10.1.2.2.2.2.2
Use the power rule to combine exponents.
Step 10.1.2.2.2.3
Add and .
Step 10.1.2.2.3
Raise to the power of .
Step 10.1.2.2.4
Multiply by .
Step 10.1.2.2.5
Raise to the power of .
Step 10.1.2.2.6
Use the Binomial Theorem.
Step 10.1.2.2.7
Simplify each term.
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Step 10.1.2.2.7.1
Raise to the power of .
Step 10.1.2.2.7.2
Raise to the power of .
Step 10.1.2.2.7.3
Multiply by .
Step 10.1.2.2.7.4
Multiply by .
Step 10.1.2.2.7.5
Multiply by .
Step 10.1.2.2.7.6
Apply the product rule to .
Step 10.1.2.2.7.7
Raise to the power of .
Step 10.1.2.2.7.8
Multiply by .
Step 10.1.2.2.7.9
Rewrite as .
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Step 10.1.2.2.7.9.1
Use to rewrite as .
Step 10.1.2.2.7.9.2
Apply the power rule and multiply exponents, .
Step 10.1.2.2.7.9.3
Combine and .
Step 10.1.2.2.7.9.4
Cancel the common factor of .
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Step 10.1.2.2.7.9.4.1
Cancel the common factor.
Step 10.1.2.2.7.9.4.2
Rewrite the expression.
Step 10.1.2.2.7.9.5
Evaluate the exponent.
Step 10.1.2.2.7.10
Multiply by .
Step 10.1.2.2.7.11
Apply the product rule to .
Step 10.1.2.2.7.12
Raise to the power of .
Step 10.1.2.2.7.13
Rewrite as .
Step 10.1.2.2.7.14
Raise to the power of .
Step 10.1.2.2.7.15
Rewrite as .
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Step 10.1.2.2.7.15.1
Factor out of .
Step 10.1.2.2.7.15.2
Rewrite as .
Step 10.1.2.2.7.16
Pull terms out from under the radical.
Step 10.1.2.2.7.17
Multiply by .
Step 10.1.2.2.8
Add and .
Step 10.1.2.2.9
Subtract from .
Step 10.1.2.2.10
Use the power rule to distribute the exponent.
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Step 10.1.2.2.10.1
Apply the product rule to .
Step 10.1.2.2.10.2
Apply the product rule to .
Step 10.1.2.2.11
Raise to the power of .
Step 10.1.2.2.12
Multiply by .
Step 10.1.2.2.13
Raise to the power of .
Step 10.1.2.2.14
Rewrite as .
Step 10.1.2.2.15
Expand using the FOIL Method.
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Step 10.1.2.2.15.1
Apply the distributive property.
Step 10.1.2.2.15.2
Apply the distributive property.
Step 10.1.2.2.15.3
Apply the distributive property.
Step 10.1.2.2.16
Simplify and combine like terms.
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Step 10.1.2.2.16.1
Simplify each term.
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Step 10.1.2.2.16.1.1
Multiply by .
Step 10.1.2.2.16.1.2
Multiply by .
Step 10.1.2.2.16.1.3
Multiply by .
Step 10.1.2.2.16.1.4
Multiply .
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Step 10.1.2.2.16.1.4.1
Multiply by .
Step 10.1.2.2.16.1.4.2
Multiply by .
Step 10.1.2.2.16.1.4.3
Raise to the power of .
Step 10.1.2.2.16.1.4.4
Raise to the power of .
Step 10.1.2.2.16.1.4.5
Use the power rule to combine exponents.
Step 10.1.2.2.16.1.4.6
Add and .
Step 10.1.2.2.16.1.5
Rewrite as .
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Step 10.1.2.2.16.1.5.1
Use to rewrite as .
Step 10.1.2.2.16.1.5.2
Apply the power rule and multiply exponents, .
Step 10.1.2.2.16.1.5.3
Combine and .
Step 10.1.2.2.16.1.5.4
Cancel the common factor of .
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Step 10.1.2.2.16.1.5.4.1
Cancel the common factor.
Step 10.1.2.2.16.1.5.4.2
Rewrite the expression.
Step 10.1.2.2.16.1.5.5
Evaluate the exponent.
Step 10.1.2.2.16.2
Add and .
Step 10.1.2.2.16.3
Subtract from .
Step 10.1.2.2.17
Combine and .
Step 10.1.2.2.18
Move the negative in front of the fraction.
Step 10.1.2.2.19
Multiply .
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Step 10.1.2.2.19.1
Multiply by .
Step 10.1.2.2.19.2
Combine and .
Step 10.1.2.3
Find the common denominator.
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Step 10.1.2.3.1
Multiply by .
Step 10.1.2.3.2
Multiply by .
Step 10.1.2.3.3
Multiply by .
Step 10.1.2.3.4
Multiply by .
Step 10.1.2.3.5
Reorder the factors of .
Step 10.1.2.3.6
Multiply by .
Step 10.1.2.3.7
Multiply by .
Step 10.1.2.4
Combine the numerators over the common denominator.
Step 10.1.2.5
Simplify each term.
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Step 10.1.2.5.1
Apply the distributive property.
Step 10.1.2.5.2
Multiply by .
Step 10.1.2.5.3
Multiply by .
Step 10.1.2.5.4
Apply the distributive property.
Step 10.1.2.5.5
Multiply by .
Step 10.1.2.5.6
Multiply by .
Step 10.1.2.5.7
Apply the distributive property.
Step 10.1.2.5.8
Multiply by .
Step 10.1.2.5.9
Multiply by .
Step 10.1.2.5.10
Apply the distributive property.
Step 10.1.2.5.11
Multiply by .
Step 10.1.2.5.12
Multiply by .
Step 10.1.2.6
Simplify by adding terms.
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Step 10.1.2.6.1
Subtract from .
Step 10.1.2.6.2
Add and .
Step 10.1.2.6.3
Add and .
Step 10.1.2.6.4
Subtract from .
Step 10.2
Write the and coordinates in point form.
Step 11
These are the turning points.
Step 12