Calculus Examples

Find the Local Maxima and Minima f(x)=2(x-1)(15x^3+5x^2-7x-1)
Step 1
Find the first derivative of the function.
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Step 1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2
Differentiate using the Product Rule which states that is where and .
Step 1.3
Differentiate.
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Step 1.3.1
By the Sum Rule, the derivative of with respect to is .
Step 1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.3.4
Multiply by .
Step 1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.6
Differentiate using the Power Rule which states that is where .
Step 1.3.7
Multiply by .
Step 1.3.8
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.9
Differentiate using the Power Rule which states that is where .
Step 1.3.10
Multiply by .
Step 1.3.11
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.12
Add and .
Step 1.3.13
By the Sum Rule, the derivative of with respect to is .
Step 1.3.14
Differentiate using the Power Rule which states that is where .
Step 1.3.15
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.16
Simplify the expression.
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Step 1.3.16.1
Add and .
Step 1.3.16.2
Multiply by .
Step 1.4
Simplify.
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Step 1.4.1
Apply the distributive property.
Step 1.4.2
Apply the distributive property.
Step 1.4.3
Apply the distributive property.
Step 1.4.4
Apply the distributive property.
Step 1.4.5
Apply the distributive property.
Step 1.4.6
Apply the distributive property.
Step 1.4.7
Combine terms.
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Step 1.4.7.1
Raise to the power of .
Step 1.4.7.2
Use the power rule to combine exponents.
Step 1.4.7.3
Add and .
Step 1.4.7.4
Multiply by .
Step 1.4.7.5
Multiply by .
Step 1.4.7.6
Multiply by .
Step 1.4.7.7
Raise to the power of .
Step 1.4.7.8
Raise to the power of .
Step 1.4.7.9
Use the power rule to combine exponents.
Step 1.4.7.10
Add and .
Step 1.4.7.11
Multiply by .
Step 1.4.7.12
Multiply by .
Step 1.4.7.13
Multiply by .
Step 1.4.7.14
Add and .
Step 1.4.7.15
Move to the left of .
Step 1.4.7.16
Multiply by .
Step 1.4.7.17
Multiply by .
Step 1.4.7.18
Multiply by .
Step 1.4.7.19
Subtract from .
Step 1.4.7.20
Multiply by .
Step 1.4.7.21
Add and .
Step 1.4.7.22
Multiply by .
Step 1.4.7.23
Add and .
Step 1.4.7.24
Multiply by .
Step 1.4.7.25
Subtract from .
Step 1.4.7.26
Multiply by .
Step 1.4.7.27
Subtract from .
Step 2
Find the second derivative of the function.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Evaluate .
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Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Multiply by .
Step 2.5
Differentiate using the Constant Rule.
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Step 2.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.2
Add and .
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
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2
Differentiate using the Product Rule which states that is where and .
Step 4.1.3
Differentiate.
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Step 4.1.3.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.3
Differentiate using the Power Rule which states that is where .
Step 4.1.3.4
Multiply by .
Step 4.1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.6
Differentiate using the Power Rule which states that is where .
Step 4.1.3.7
Multiply by .
Step 4.1.3.8
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.9
Differentiate using the Power Rule which states that is where .
Step 4.1.3.10
Multiply by .
Step 4.1.3.11
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.12
Add and .
Step 4.1.3.13
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3.14
Differentiate using the Power Rule which states that is where .
Step 4.1.3.15
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.16
Simplify the expression.
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Step 4.1.3.16.1
Add and .
Step 4.1.3.16.2
Multiply by .
Step 4.1.4
Simplify.
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Step 4.1.4.1
Apply the distributive property.
Step 4.1.4.2
Apply the distributive property.
Step 4.1.4.3
Apply the distributive property.
Step 4.1.4.4
Apply the distributive property.
Step 4.1.4.5
Apply the distributive property.
Step 4.1.4.6
Apply the distributive property.
Step 4.1.4.7
Combine terms.
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Step 4.1.4.7.1
Raise to the power of .
Step 4.1.4.7.2
Use the power rule to combine exponents.
Step 4.1.4.7.3
Add and .
Step 4.1.4.7.4
Multiply by .
Step 4.1.4.7.5
Multiply by .
Step 4.1.4.7.6
Multiply by .
Step 4.1.4.7.7
Raise to the power of .
Step 4.1.4.7.8
Raise to the power of .
Step 4.1.4.7.9
Use the power rule to combine exponents.
Step 4.1.4.7.10
Add and .
Step 4.1.4.7.11
Multiply by .
Step 4.1.4.7.12
Multiply by .
Step 4.1.4.7.13
Multiply by .
Step 4.1.4.7.14
Add and .
Step 4.1.4.7.15
Move to the left of .
Step 4.1.4.7.16
Multiply by .
Step 4.1.4.7.17
Multiply by .
Step 4.1.4.7.18
Multiply by .
Step 4.1.4.7.19
Subtract from .
Step 4.1.4.7.20
Multiply by .
Step 4.1.4.7.21
Add and .
Step 4.1.4.7.22
Multiply by .
Step 4.1.4.7.23
Add and .
Step 4.1.4.7.24
Multiply by .
Step 4.1.4.7.25
Subtract from .
Step 4.1.4.7.26
Multiply by .
Step 4.1.4.7.27
Subtract from .
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
Graph each side of the equation. The solution is the x-value of the point of intersection.
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 each term.
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Step 9.1.1
Raise to the power of .
Step 9.1.2
Multiply by .
Step 9.1.3
Multiply by .
Step 9.2
Simplify by adding and subtracting.
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Step 9.2.1
Add and .
Step 9.2.2
Subtract from .
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
Simplify the expression.
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Step 11.2.1.1
Subtract from .
Step 11.2.1.2
Multiply by .
Step 11.2.2
Simplify each term.
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Step 11.2.2.1
Raise to the power of .
Step 11.2.2.2
Multiply by .
Step 11.2.2.3
Raise to the power of .
Step 11.2.2.4
Multiply by .
Step 11.2.2.5
Multiply by .
Step 11.2.3
Simplify the expression.
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Step 11.2.3.1
Add and .
Step 11.2.3.2
Add and .
Step 11.2.3.3
Subtract from .
Step 11.2.3.4
Multiply 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 each term.
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Step 13.1.1
Raise to the power of .
Step 13.1.2
Multiply by .
Step 13.1.3
Multiply by .
Step 13.2
Simplify by subtracting numbers.
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Step 13.2.1
Subtract from .
Step 13.2.2
Subtract from .
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
Simplify the expression.
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Step 15.2.1.1
Subtract from .
Step 15.2.1.2
Multiply by .
Step 15.2.2
Simplify each term.
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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
Raise to the power of .
Step 15.2.2.4
Multiply by .
Step 15.2.2.5
Multiply by .
Step 15.2.3
Simplify the expression.
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Step 15.2.3.1
Add and .
Step 15.2.3.2
Subtract from .
Step 15.2.3.3
Subtract from .
Step 15.2.3.4
Multiply by .
Step 15.2.4
The final answer is .
Step 16
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 17
Evaluate the second derivative.
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Step 17.1
Simplify each term.
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Step 17.1.1
Raise to the power of .
Step 17.1.2
Multiply by .
Step 17.1.3
Multiply by .
Step 17.2
Simplify by subtracting numbers.
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Step 17.2.1
Subtract from .
Step 17.2.2
Subtract from .
Step 18
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 19
Find the y-value when .
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Step 19.1
Replace the variable with in the expression.
Step 19.2
Simplify the result.
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Step 19.2.1
Simplify the expression.
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Step 19.2.1.1
Subtract from .
Step 19.2.1.2
Multiply by .
Step 19.2.2
Simplify each term.
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Step 19.2.2.1
Raise to the power of .
Step 19.2.2.2
Multiply by .
Step 19.2.2.3
Raise to the power of .
Step 19.2.2.4
Multiply by .
Step 19.2.2.5
Multiply by .
Step 19.2.3
Simplify the expression.
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Step 19.2.3.1
Add and .
Step 19.2.3.2
Subtract from .
Step 19.2.3.3
Subtract from .
Step 19.2.3.4
Multiply by .
Step 19.2.4
The final answer is .
Step 20
These are the local extrema for .
is a local minima
is a local maxima
is a local minima
Step 21