Calculus Examples

Find the Critical Points f(x)=2x^3-10x^2-6x+1
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Evaluate .
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Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Multiply by .
Step 1.1.3
Evaluate .
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Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Evaluate .
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Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.1.4.3
Multiply by .
Step 1.1.5
Differentiate using the Constant Rule.
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Step 1.1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.5.2
Add and .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Factor out of .
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Step 2.2.1
Factor out of .
Step 2.2.2
Factor out of .
Step 2.2.3
Factor out of .
Step 2.2.4
Factor out of .
Step 2.2.5
Factor out of .
Step 2.3
Divide each term in by and simplify.
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Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Cancel the common factor of .
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Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
Divide by .
Step 2.4
Use the quadratic formula to find the solutions.
Step 2.5
Substitute the values , , and into the quadratic formula and solve for .
Step 2.6
Simplify.
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Step 2.6.1
Simplify the numerator.
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Step 2.6.1.1
Raise to the power of .
Step 2.6.1.2
Multiply .
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Step 2.6.1.2.1
Multiply by .
Step 2.6.1.2.2
Multiply by .
Step 2.6.1.3
Add and .
Step 2.6.1.4
Rewrite as .
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Step 2.6.1.4.1
Factor out of .
Step 2.6.1.4.2
Rewrite as .
Step 2.6.1.5
Pull terms out from under the radical.
Step 2.6.2
Multiply by .
Step 2.6.3
Simplify .
Step 2.7
Simplify the expression to solve for the portion of the .
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Step 2.7.1
Simplify the numerator.
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Step 2.7.1.1
Raise to the power of .
Step 2.7.1.2
Multiply .
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Step 2.7.1.2.1
Multiply by .
Step 2.7.1.2.2
Multiply by .
Step 2.7.1.3
Add and .
Step 2.7.1.4
Rewrite as .
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Step 2.7.1.4.1
Factor out of .
Step 2.7.1.4.2
Rewrite as .
Step 2.7.1.5
Pull terms out from under the radical.
Step 2.7.2
Multiply by .
Step 2.7.3
Simplify .
Step 2.7.4
Change the to .
Step 2.8
Simplify the expression to solve for the portion of the .
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Step 2.8.1
Simplify the numerator.
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Step 2.8.1.1
Raise to the power of .
Step 2.8.1.2
Multiply .
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Step 2.8.1.2.1
Multiply by .
Step 2.8.1.2.2
Multiply by .
Step 2.8.1.3
Add and .
Step 2.8.1.4
Rewrite as .
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Step 2.8.1.4.1
Factor out of .
Step 2.8.1.4.2
Rewrite as .
Step 2.8.1.5
Pull terms out from under the radical.
Step 2.8.2
Multiply by .
Step 2.8.3
Simplify .
Step 2.8.4
Change the to .
Step 2.9
The final answer is the combination of both solutions.
Step 3
Find the values where the derivative is undefined.
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Step 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 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
Simplify each term.
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Step 4.1.2.1.1
Apply the product rule to .
Step 4.1.2.1.2
Raise to the power of .
Step 4.1.2.1.3
Use the Binomial Theorem.
Step 4.1.2.1.4
Simplify each term.
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Step 4.1.2.1.4.1
Raise to the power of .
Step 4.1.2.1.4.2
Raise to the power of .
Step 4.1.2.1.4.3
Multiply by .
Step 4.1.2.1.4.4
Multiply by .
Step 4.1.2.1.4.5
Rewrite as .
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Step 4.1.2.1.4.5.1
Use to rewrite as .
Step 4.1.2.1.4.5.2
Apply the power rule and multiply exponents, .
Step 4.1.2.1.4.5.3
Combine and .
Step 4.1.2.1.4.5.4
Cancel the common factor of .
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Step 4.1.2.1.4.5.4.1
Cancel the common factor.
Step 4.1.2.1.4.5.4.2
Rewrite the expression.
Step 4.1.2.1.4.5.5
Evaluate the exponent.
Step 4.1.2.1.4.6
Multiply by .
Step 4.1.2.1.4.7
Rewrite as .
Step 4.1.2.1.4.8
Raise to the power of .
Step 4.1.2.1.4.9
Rewrite as .
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Step 4.1.2.1.4.9.1
Factor out of .
Step 4.1.2.1.4.9.2
Rewrite as .
Step 4.1.2.1.4.10
Pull terms out from under the radical.
Step 4.1.2.1.5
Add and .
Step 4.1.2.1.6
Add and .
Step 4.1.2.1.7
Combine and .
Step 4.1.2.1.8
Apply the product rule to .
Step 4.1.2.1.9
Raise to the power of .
Step 4.1.2.1.10
Rewrite as .
Step 4.1.2.1.11
Expand using the FOIL Method.
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Step 4.1.2.1.11.1
Apply the distributive property.
Step 4.1.2.1.11.2
Apply the distributive property.
Step 4.1.2.1.11.3
Apply the distributive property.
Step 4.1.2.1.12
Simplify and combine like terms.
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Step 4.1.2.1.12.1
Simplify each term.
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Step 4.1.2.1.12.1.1
Multiply by .
Step 4.1.2.1.12.1.2
Move to the left of .
Step 4.1.2.1.12.1.3
Combine using the product rule for radicals.
Step 4.1.2.1.12.1.4
Multiply by .
Step 4.1.2.1.12.1.5
Rewrite as .
Step 4.1.2.1.12.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 4.1.2.1.12.2
Add and .
Step 4.1.2.1.12.3
Add and .
Step 4.1.2.1.13
Combine and .
Step 4.1.2.1.14
Move the negative in front of the fraction.
Step 4.1.2.1.15
Cancel the common factor of .
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Step 4.1.2.1.15.1
Factor out of .
Step 4.1.2.1.15.2
Cancel the common factor.
Step 4.1.2.1.15.3
Rewrite the expression.
Step 4.1.2.1.16
Apply the distributive property.
Step 4.1.2.1.17
Multiply by .
Step 4.1.2.2
To write as a fraction with a common denominator, multiply by .
Step 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 4.1.2.3.1
Multiply by .
Step 4.1.2.3.2
Multiply by .
Step 4.1.2.4
Combine the numerators over the common denominator.
Step 4.1.2.5
Simplify the numerator.
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Step 4.1.2.5.1
Apply the distributive property.
Step 4.1.2.5.2
Multiply by .
Step 4.1.2.5.3
Multiply by .
Step 4.1.2.5.4
Apply the distributive property.
Step 4.1.2.5.5
Multiply by .
Step 4.1.2.5.6
Multiply by .
Step 4.1.2.5.7
Apply the distributive property.
Step 4.1.2.5.8
Multiply by .
Step 4.1.2.5.9
Multiply by .
Step 4.1.2.5.10
Subtract from .
Step 4.1.2.5.11
Subtract from .
Step 4.1.2.6
To write as a fraction with a common denominator, multiply by .
Step 4.1.2.7
Combine and .
Step 4.1.2.8
Simplify the expression.
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Step 4.1.2.8.1
Combine the numerators over the common denominator.
Step 4.1.2.8.2
Multiply by .
Step 4.1.2.8.3
Subtract from .
Step 4.1.2.9
To write as a fraction with a common denominator, multiply by .
Step 4.1.2.10
Combine fractions.
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Step 4.1.2.10.1
Combine and .
Step 4.1.2.10.2
Combine the numerators over the common denominator.
Step 4.1.2.11
Simplify the numerator.
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Step 4.1.2.11.1
Multiply by .
Step 4.1.2.11.2
Subtract from .
Step 4.1.2.12
Simplify with factoring out.
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Step 4.1.2.12.1
Write as a fraction with a common denominator.
Step 4.1.2.12.2
Simplify the expression.
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Step 4.1.2.12.2.1
Combine the numerators over the common denominator.
Step 4.1.2.12.2.2
Add and .
Step 4.1.2.12.3
Rewrite as .
Step 4.1.2.12.4
Factor out of .
Step 4.1.2.12.5
Factor out of .
Step 4.1.2.12.6
Move the negative in front of the fraction.
Step 4.2
Evaluate at .
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Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
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Step 4.2.2.1
Simplify each term.
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Step 4.2.2.1.1
Apply the product rule to .
Step 4.2.2.1.2
Raise to the power of .
Step 4.2.2.1.3
Use the Binomial Theorem.
Step 4.2.2.1.4
Simplify each term.
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Step 4.2.2.1.4.1
Raise to the power of .
Step 4.2.2.1.4.2
Raise to the power of .
Step 4.2.2.1.4.3
Multiply by .
Step 4.2.2.1.4.4
Multiply by .
Step 4.2.2.1.4.5
Multiply by .
Step 4.2.2.1.4.6
Apply the product rule to .
Step 4.2.2.1.4.7
Raise to the power of .
Step 4.2.2.1.4.8
Multiply by .
Step 4.2.2.1.4.9
Rewrite as .
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Step 4.2.2.1.4.9.1
Use to rewrite as .
Step 4.2.2.1.4.9.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.4.9.3
Combine and .
Step 4.2.2.1.4.9.4
Cancel the common factor of .
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Step 4.2.2.1.4.9.4.1
Cancel the common factor.
Step 4.2.2.1.4.9.4.2
Rewrite the expression.
Step 4.2.2.1.4.9.5
Evaluate the exponent.
Step 4.2.2.1.4.10
Multiply by .
Step 4.2.2.1.4.11
Apply the product rule to .
Step 4.2.2.1.4.12
Raise to the power of .
Step 4.2.2.1.4.13
Rewrite as .
Step 4.2.2.1.4.14
Raise to the power of .
Step 4.2.2.1.4.15
Rewrite as .
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Step 4.2.2.1.4.15.1
Factor out of .
Step 4.2.2.1.4.15.2
Rewrite as .
Step 4.2.2.1.4.16
Pull terms out from under the radical.
Step 4.2.2.1.4.17
Multiply by .
Step 4.2.2.1.5
Add and .
Step 4.2.2.1.6
Subtract from .
Step 4.2.2.1.7
Combine and .
Step 4.2.2.1.8
Apply the product rule to .
Step 4.2.2.1.9
Raise to the power of .
Step 4.2.2.1.10
Rewrite as .
Step 4.2.2.1.11
Expand using the FOIL Method.
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Step 4.2.2.1.11.1
Apply the distributive property.
Step 4.2.2.1.11.2
Apply the distributive property.
Step 4.2.2.1.11.3
Apply the distributive property.
Step 4.2.2.1.12
Simplify and combine like terms.
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Step 4.2.2.1.12.1
Simplify each term.
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Step 4.2.2.1.12.1.1
Multiply by .
Step 4.2.2.1.12.1.2
Multiply by .
Step 4.2.2.1.12.1.3
Multiply by .
Step 4.2.2.1.12.1.4
Multiply .
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Step 4.2.2.1.12.1.4.1
Multiply by .
Step 4.2.2.1.12.1.4.2
Multiply by .
Step 4.2.2.1.12.1.4.3
Raise to the power of .
Step 4.2.2.1.12.1.4.4
Raise to the power of .
Step 4.2.2.1.12.1.4.5
Use the power rule to combine exponents.
Step 4.2.2.1.12.1.4.6
Add and .
Step 4.2.2.1.12.1.5
Rewrite as .
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Step 4.2.2.1.12.1.5.1
Use to rewrite as .
Step 4.2.2.1.12.1.5.2
Apply the power rule and multiply exponents, .
Step 4.2.2.1.12.1.5.3
Combine and .
Step 4.2.2.1.12.1.5.4
Cancel the common factor of .
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Step 4.2.2.1.12.1.5.4.1
Cancel the common factor.
Step 4.2.2.1.12.1.5.4.2
Rewrite the expression.
Step 4.2.2.1.12.1.5.5
Evaluate the exponent.
Step 4.2.2.1.12.2
Add and .
Step 4.2.2.1.12.3
Subtract from .
Step 4.2.2.1.13
Combine and .
Step 4.2.2.1.14
Move the negative in front of the fraction.
Step 4.2.2.1.15
Cancel the common factor of .
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Step 4.2.2.1.15.1
Factor out of .
Step 4.2.2.1.15.2
Cancel the common factor.
Step 4.2.2.1.15.3
Rewrite the expression.
Step 4.2.2.1.16
Apply the distributive property.
Step 4.2.2.1.17
Multiply by .
Step 4.2.2.1.18
Multiply by .
Step 4.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 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 4.2.2.3.1
Multiply by .
Step 4.2.2.3.2
Multiply by .
Step 4.2.2.4
Combine the numerators over the common denominator.
Step 4.2.2.5
Simplify the numerator.
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Step 4.2.2.5.1
Apply the distributive property.
Step 4.2.2.5.2
Multiply by .
Step 4.2.2.5.3
Multiply by .
Step 4.2.2.5.4
Apply the distributive property.
Step 4.2.2.5.5
Multiply by .
Step 4.2.2.5.6
Multiply by .
Step 4.2.2.5.7
Apply the distributive property.
Step 4.2.2.5.8
Multiply by .
Step 4.2.2.5.9
Multiply by .
Step 4.2.2.5.10
Subtract from .
Step 4.2.2.5.11
Add and .
Step 4.2.2.6
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.7
Combine and .
Step 4.2.2.8
Simplify the expression.
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Step 4.2.2.8.1
Combine the numerators over the common denominator.
Step 4.2.2.8.2
Multiply by .
Step 4.2.2.8.3
Subtract from .
Step 4.2.2.9
To write as a fraction with a common denominator, multiply by .
Step 4.2.2.10
Combine fractions.
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Step 4.2.2.10.1
Combine and .
Step 4.2.2.10.2
Combine the numerators over the common denominator.
Step 4.2.2.11
Simplify the numerator.
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Step 4.2.2.11.1
Multiply by .
Step 4.2.2.11.2
Add and .
Step 4.2.2.12
Simplify with factoring out.
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Step 4.2.2.12.1
Write as a fraction with a common denominator.
Step 4.2.2.12.2
Simplify the expression.
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Step 4.2.2.12.2.1
Combine the numerators over the common denominator.
Step 4.2.2.12.2.2
Add and .
Step 4.2.2.12.3
Rewrite as .
Step 4.2.2.12.4
Factor out of .
Step 4.2.2.12.5
Factor out of .
Step 4.2.2.12.6
Move the negative in front of the fraction.
Step 4.3
List all of the points.
Step 5