Calculus Examples

Find the Local Maxima and Minima g(y)=(y-1)/(y^2-y+1)
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
Differentiate.
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Step 1.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.4
Simplify the expression.
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Step 1.2.4.1
Add and .
Step 1.2.4.2
Multiply by .
Step 1.2.5
By the Sum Rule, the derivative of with respect to is .
Step 1.2.6
Differentiate using the Power Rule which states that is where .
Step 1.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.8
Differentiate using the Power Rule which states that is where .
Step 1.2.9
Multiply by .
Step 1.2.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.11
Add and .
Step 1.3
Simplify.
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Step 1.3.1
Apply the distributive property.
Step 1.3.2
Simplify the numerator.
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Step 1.3.2.1
Simplify each term.
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Step 1.3.2.1.1
Multiply by .
Step 1.3.2.1.2
Expand using the FOIL Method.
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Step 1.3.2.1.2.1
Apply the distributive property.
Step 1.3.2.1.2.2
Apply the distributive property.
Step 1.3.2.1.2.3
Apply the distributive property.
Step 1.3.2.1.3
Simplify and combine like terms.
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Step 1.3.2.1.3.1
Simplify each term.
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Step 1.3.2.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 1.3.2.1.3.1.2
Multiply by by adding the exponents.
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Step 1.3.2.1.3.1.2.1
Move .
Step 1.3.2.1.3.1.2.2
Multiply by .
Step 1.3.2.1.3.1.3
Multiply by .
Step 1.3.2.1.3.1.4
Multiply .
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Step 1.3.2.1.3.1.4.1
Multiply by .
Step 1.3.2.1.3.1.4.2
Multiply by .
Step 1.3.2.1.3.1.5
Multiply by .
Step 1.3.2.1.3.1.6
Multiply by .
Step 1.3.2.1.3.2
Add and .
Step 1.3.2.2
Combine the opposite terms in .
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Step 1.3.2.2.1
Subtract from .
Step 1.3.2.2.2
Add and .
Step 1.3.2.3
Subtract from .
Step 1.3.2.4
Add and .
Step 1.3.3
Factor out of .
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Step 1.3.3.1
Factor out of .
Step 1.3.3.2
Factor out of .
Step 1.3.3.3
Factor out of .
Step 1.3.4
Factor out of .
Step 1.3.5
Rewrite as .
Step 1.3.6
Factor out of .
Step 1.3.7
Rewrite as .
Step 1.3.8
Move the negative in front of the fraction.
Step 2
Find the second derivative of the function.
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Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Multiply the exponents in .
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Step 2.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2
Multiply by .
Step 2.4
Differentiate using the Product Rule which states that is where and .
Step 2.5
Differentiate.
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Step 2.5.1
By the Sum Rule, the derivative of with respect to is .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.4
Simplify the expression.
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Step 2.5.4.1
Add and .
Step 2.5.4.2
Multiply by .
Step 2.5.5
Differentiate using the Power Rule which states that is where .
Step 2.5.6
Simplify by adding terms.
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Step 2.5.6.1
Multiply by .
Step 2.5.6.2
Add and .
Step 2.6
Differentiate using the chain rule, which states that is where and .
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Step 2.6.1
To apply the Chain Rule, set as .
Step 2.6.2
Differentiate using the Power Rule which states that is where .
Step 2.6.3
Replace all occurrences of with .
Step 2.7
Simplify with factoring out.
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Step 2.7.1
Multiply by .
Step 2.7.2
Factor out of .
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Step 2.7.2.1
Factor out of .
Step 2.7.2.2
Factor out of .
Step 2.7.2.3
Factor out of .
Step 2.8
Cancel the common factors.
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Step 2.8.1
Factor out of .
Step 2.8.2
Cancel the common factor.
Step 2.8.3
Rewrite the expression.
Step 2.9
By the Sum Rule, the derivative of with respect to is .
Step 2.10
Differentiate using the Power Rule which states that is where .
Step 2.11
Since is constant with respect to , the derivative of with respect to is .
Step 2.12
Differentiate using the Power Rule which states that is where .
Step 2.13
Multiply by .
Step 2.14
Since is constant with respect to , the derivative of with respect to is .
Step 2.15
Add and .
Step 2.16
Since is constant with respect to , the derivative of with respect to is .
Step 2.17
Simplify the expression.
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Step 2.17.1
Multiply by .
Step 2.17.2
Add and .
Step 2.18
Simplify.
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Step 2.18.1
Apply the distributive property.
Step 2.18.2
Simplify the numerator.
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Step 2.18.2.1
Simplify each term.
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Step 2.18.2.1.1
Expand by multiplying each term in the first expression by each term in the second expression.
Step 2.18.2.1.2
Simplify each term.
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Step 2.18.2.1.2.1
Rewrite using the commutative property of multiplication.
Step 2.18.2.1.2.2
Multiply by by adding the exponents.
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Step 2.18.2.1.2.2.1
Move .
Step 2.18.2.1.2.2.2
Multiply by .
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Step 2.18.2.1.2.2.2.1
Raise to the power of .
Step 2.18.2.1.2.2.2.2
Use the power rule to combine exponents.
Step 2.18.2.1.2.2.3
Add and .
Step 2.18.2.1.2.3
Move to the left of .
Step 2.18.2.1.2.4
Rewrite using the commutative property of multiplication.
Step 2.18.2.1.2.5
Multiply by by adding the exponents.
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Step 2.18.2.1.2.5.1
Move .
Step 2.18.2.1.2.5.2
Multiply by .
Step 2.18.2.1.2.6
Multiply by .
Step 2.18.2.1.2.7
Multiply by .
Step 2.18.2.1.2.8
Multiply by .
Step 2.18.2.1.2.9
Multiply by .
Step 2.18.2.1.3
Subtract from .
Step 2.18.2.1.4
Add and .
Step 2.18.2.1.5
Simplify each term.
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Step 2.18.2.1.5.1
Multiply by by adding the exponents.
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Step 2.18.2.1.5.1.1
Move .
Step 2.18.2.1.5.1.2
Multiply by .
Step 2.18.2.1.5.2
Multiply by .
Step 2.18.2.1.6
Expand using the FOIL Method.
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Step 2.18.2.1.6.1
Apply the distributive property.
Step 2.18.2.1.6.2
Apply the distributive property.
Step 2.18.2.1.6.3
Apply the distributive property.
Step 2.18.2.1.7
Simplify and combine like terms.
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Step 2.18.2.1.7.1
Simplify each term.
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Step 2.18.2.1.7.1.1
Rewrite using the commutative property of multiplication.
Step 2.18.2.1.7.1.2
Multiply by by adding the exponents.
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Step 2.18.2.1.7.1.2.1
Move .
Step 2.18.2.1.7.1.2.2
Multiply by .
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Step 2.18.2.1.7.1.2.2.1
Raise to the power of .
Step 2.18.2.1.7.1.2.2.2
Use the power rule to combine exponents.
Step 2.18.2.1.7.1.2.3
Add and .
Step 2.18.2.1.7.1.3
Multiply by .
Step 2.18.2.1.7.1.4
Multiply by .
Step 2.18.2.1.7.1.5
Rewrite using the commutative property of multiplication.
Step 2.18.2.1.7.1.6
Multiply by by adding the exponents.
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Step 2.18.2.1.7.1.6.1
Move .
Step 2.18.2.1.7.1.6.2
Multiply by .
Step 2.18.2.1.7.1.7
Multiply by .
Step 2.18.2.1.7.1.8
Multiply by .
Step 2.18.2.1.7.2
Add and .
Step 2.18.2.2
Combine the opposite terms in .
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Step 2.18.2.2.1
Subtract from .
Step 2.18.2.2.2
Add and .
Step 2.18.2.3
Subtract from .
Step 2.18.2.4
Add and .
Step 2.18.3
Factor out of .
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Step 2.18.3.1
Factor out of .
Step 2.18.3.2
Factor out of .
Step 2.18.3.3
Factor out of .
Step 2.18.3.4
Factor out of .
Step 2.18.3.5
Factor out of .
Step 2.18.4
Factor out of .
Step 2.18.5
Factor out of .
Step 2.18.6
Factor out of .
Step 2.18.7
Rewrite as .
Step 2.18.8
Factor out of .
Step 2.18.9
Rewrite as .
Step 2.18.10
Move the negative in front of the fraction.
Step 2.18.11
Multiply by .
Step 2.18.12
Multiply by .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Find the first derivative.
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Step 4.1
Find the first derivative.
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Step 4.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 4.1.2
Differentiate.
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Step 4.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.4
Simplify the expression.
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Step 4.1.2.4.1
Add and .
Step 4.1.2.4.2
Multiply by .
Step 4.1.2.5
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2.6
Differentiate using the Power Rule which states that is where .
Step 4.1.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.8
Differentiate using the Power Rule which states that is where .
Step 4.1.2.9
Multiply by .
Step 4.1.2.10
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.11
Add and .
Step 4.1.3
Simplify.
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Step 4.1.3.1
Apply the distributive property.
Step 4.1.3.2
Simplify the numerator.
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Step 4.1.3.2.1
Simplify each term.
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Step 4.1.3.2.1.1
Multiply by .
Step 4.1.3.2.1.2
Expand using the FOIL Method.
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Step 4.1.3.2.1.2.1
Apply the distributive property.
Step 4.1.3.2.1.2.2
Apply the distributive property.
Step 4.1.3.2.1.2.3
Apply the distributive property.
Step 4.1.3.2.1.3
Simplify and combine like terms.
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Step 4.1.3.2.1.3.1
Simplify each term.
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Step 4.1.3.2.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 4.1.3.2.1.3.1.2
Multiply by by adding the exponents.
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Step 4.1.3.2.1.3.1.2.1
Move .
Step 4.1.3.2.1.3.1.2.2
Multiply by .
Step 4.1.3.2.1.3.1.3
Multiply by .
Step 4.1.3.2.1.3.1.4
Multiply .
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Step 4.1.3.2.1.3.1.4.1
Multiply by .
Step 4.1.3.2.1.3.1.4.2
Multiply by .
Step 4.1.3.2.1.3.1.5
Multiply by .
Step 4.1.3.2.1.3.1.6
Multiply by .
Step 4.1.3.2.1.3.2
Add and .
Step 4.1.3.2.2
Combine the opposite terms in .
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Step 4.1.3.2.2.1
Subtract from .
Step 4.1.3.2.2.2
Add and .
Step 4.1.3.2.3
Subtract from .
Step 4.1.3.2.4
Add and .
Step 4.1.3.3
Factor out of .
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Step 4.1.3.3.1
Factor out of .
Step 4.1.3.3.2
Factor out of .
Step 4.1.3.3.3
Factor out of .
Step 4.1.3.4
Factor out of .
Step 4.1.3.5
Rewrite as .
Step 4.1.3.6
Factor out of .
Step 4.1.3.7
Rewrite as .
Step 4.1.3.8
Move the negative in front of the fraction.
Step 4.2
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
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Step 5.1
Set the first derivative equal to .
Step 5.2
Set the numerator equal to zero.
Step 5.3
Solve the equation for .
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Step 5.3.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.3.2
Set equal to .
Step 5.3.3
Set equal to and solve for .
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Step 5.3.3.1
Set equal to .
Step 5.3.3.2
Add to both sides of the equation.
Step 5.3.4
The final solution is all the values that make true.
Step 6
Find the values where the derivative is undefined.
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Step 6.1
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 7
Critical points to evaluate.
Step 8
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 9
Evaluate the second derivative.
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Step 9.1
Simplify the numerator.
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Step 9.1.1
Raising to any positive power yields .
Step 9.1.2
Raising to any positive power yields .
Step 9.1.3
Multiply by .
Step 9.1.4
Add and .
Step 9.1.5
Add and .
Step 9.2
Simplify the denominator.
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Step 9.2.1
Raising to any positive power yields .
Step 9.2.2
Multiply by .
Step 9.2.3
Add and .
Step 9.2.4
Add and .
Step 9.2.5
One to any power is one.
Step 9.3
Simplify the expression.
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Step 9.3.1
Multiply by .
Step 9.3.2
Divide by .
Step 10
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 11
Find the y-value when .
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Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
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Step 11.2.1
Subtract from .
Step 11.2.2
Simplify the denominator.
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Step 11.2.2.1
Raising to any positive power yields .
Step 11.2.2.2
Multiply by .
Step 11.2.2.3
Add and .
Step 11.2.2.4
Add and .
Step 11.2.3
Divide by .
Step 11.2.4
The final answer is .
Step 12
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 13
Evaluate the second derivative.
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Step 13.1
Simplify the numerator.
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Step 13.1.1
Raise to the power of .
Step 13.1.2
Raise to the power of .
Step 13.1.3
Multiply by .
Step 13.1.4
Subtract from .
Step 13.1.5
Add and .
Step 13.2
Simplify the denominator.
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Step 13.2.1
Raise to the power of .
Step 13.2.2
Multiply by .
Step 13.2.3
Subtract from .
Step 13.2.4
Add and .
Step 13.2.5
Raise to the power of .
Step 13.3
Reduce the expression by cancelling the common factors.
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Step 13.3.1
Multiply by .
Step 13.3.2
Cancel the common factor of and .
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Step 13.3.2.1
Factor out of .
Step 13.3.2.2
Cancel the common factors.
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Step 13.3.2.2.1
Factor out of .
Step 13.3.2.2.2
Cancel the common factor.
Step 13.3.2.2.3
Rewrite the expression.
Step 13.3.3
Move the negative in front of the fraction.
Step 14
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 15
Find the y-value when .
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Step 15.1
Replace the variable with in the expression.
Step 15.2
Simplify the result.
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Step 15.2.1
Subtract from .
Step 15.2.2
Simplify the denominator.
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Step 15.2.2.1
Raise to the power of .
Step 15.2.2.2
Multiply by .
Step 15.2.2.3
Subtract from .
Step 15.2.2.4
Add and .
Step 15.2.3
The final answer is .
Step 16
These are the local extrema for .
is a local minima
is a local maxima
Step 17