Calculus Examples

Find the Local Maxima and Minima f(x)=(x-2)(x-3)^2
Step 1
Find the first derivative of the function.
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Step 1.1
Rewrite as .
Step 1.2
Expand using the FOIL Method.
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Step 1.2.1
Apply the distributive property.
Step 1.2.2
Apply the distributive property.
Step 1.2.3
Apply the distributive property.
Step 1.3
Simplify and combine like terms.
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Step 1.3.1
Simplify each term.
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Step 1.3.1.1
Multiply by .
Step 1.3.1.2
Move to the left of .
Step 1.3.1.3
Multiply by .
Step 1.3.2
Subtract from .
Step 1.4
Differentiate using the Product Rule which states that is where and .
Step 1.5
Differentiate.
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Step 1.5.1
By the Sum Rule, the derivative of with respect to is .
Step 1.5.2
Differentiate using the Power Rule which states that is where .
Step 1.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.5.4
Differentiate using the Power Rule which states that is where .
Step 1.5.5
Multiply by .
Step 1.5.6
Since is constant with respect to , the derivative of with respect to is .
Step 1.5.7
Add and .
Step 1.5.8
By the Sum Rule, the derivative of with respect to is .
Step 1.5.9
Differentiate using the Power Rule which states that is where .
Step 1.5.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.5.11
Simplify the expression.
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Step 1.5.11.1
Add and .
Step 1.5.11.2
Multiply by .
Step 1.6
Simplify.
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Step 1.6.1
Apply the distributive property.
Step 1.6.2
Apply the distributive property.
Step 1.6.3
Apply the distributive property.
Step 1.6.4
Combine terms.
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Step 1.6.4.1
Raise to the power of .
Step 1.6.4.2
Raise to the power of .
Step 1.6.4.3
Use the power rule to combine exponents.
Step 1.6.4.4
Add and .
Step 1.6.4.5
Multiply by .
Step 1.6.4.6
Move to the left of .
Step 1.6.4.7
Multiply by .
Step 1.6.4.8
Subtract from .
Step 1.6.4.9
Add and .
Step 1.6.4.10
Subtract from .
Step 1.6.4.11
Add and .
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
Differentiate using the Constant Rule.
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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
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
Rewrite as .
Step 4.1.2
Expand using the FOIL Method.
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Step 4.1.2.1
Apply the distributive property.
Step 4.1.2.2
Apply the distributive property.
Step 4.1.2.3
Apply the distributive property.
Step 4.1.3
Simplify and combine like terms.
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Step 4.1.3.1
Simplify each term.
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Step 4.1.3.1.1
Multiply by .
Step 4.1.3.1.2
Move to the left of .
Step 4.1.3.1.3
Multiply by .
Step 4.1.3.2
Subtract from .
Step 4.1.4
Differentiate using the Product Rule which states that is where and .
Step 4.1.5
Differentiate.
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Step 4.1.5.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.5.2
Differentiate using the Power Rule which states that is where .
Step 4.1.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5.4
Differentiate using the Power Rule which states that is where .
Step 4.1.5.5
Multiply by .
Step 4.1.5.6
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5.7
Add and .
Step 4.1.5.8
By the Sum Rule, the derivative of with respect to is .
Step 4.1.5.9
Differentiate using the Power Rule which states that is where .
Step 4.1.5.10
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5.11
Simplify the expression.
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Step 4.1.5.11.1
Add and .
Step 4.1.5.11.2
Multiply by .
Step 4.1.6
Simplify.
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Step 4.1.6.1
Apply the distributive property.
Step 4.1.6.2
Apply the distributive property.
Step 4.1.6.3
Apply the distributive property.
Step 4.1.6.4
Combine terms.
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Step 4.1.6.4.1
Raise to the power of .
Step 4.1.6.4.2
Raise to the power of .
Step 4.1.6.4.3
Use the power rule to combine exponents.
Step 4.1.6.4.4
Add and .
Step 4.1.6.4.5
Multiply by .
Step 4.1.6.4.6
Move to the left of .
Step 4.1.6.4.7
Multiply by .
Step 4.1.6.4.8
Subtract from .
Step 4.1.6.4.9
Add and .
Step 4.1.6.4.10
Subtract from .
Step 4.1.6.4.11
Add and .
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
Factor by grouping.
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Step 5.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 5.2.1.1
Factor out of .
Step 5.2.1.2
Rewrite as plus
Step 5.2.1.3
Apply the distributive property.
Step 5.2.2
Factor out the greatest common factor from each group.
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Step 5.2.2.1
Group the first two terms and the last two terms.
Step 5.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4
Set equal to and solve for .
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Step 5.4.1
Set equal to .
Step 5.4.2
Solve for .
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Step 5.4.2.1
Add to both sides of the equation.
Step 5.4.2.2
Divide each term in by and simplify.
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Step 5.4.2.2.1
Divide each term in by .
Step 5.4.2.2.2
Simplify the left side.
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Step 5.4.2.2.2.1
Cancel the common factor of .
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Step 5.4.2.2.2.1.1
Cancel the common factor.
Step 5.4.2.2.2.1.2
Divide by .
Step 5.5
Set equal to and solve for .
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Step 5.5.1
Set equal to .
Step 5.5.2
Add to both sides of the equation.
Step 5.6
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 each term.
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Step 9.1.1
Cancel the common factor of .
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Step 9.1.1.1
Factor out of .
Step 9.1.1.2
Cancel the common factor.
Step 9.1.1.3
Rewrite the expression.
Step 9.1.2
Multiply by .
Step 9.2
Subtract from .
Step 10
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 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
To write as a fraction with a common denominator, multiply by .
Step 11.2.2
Combine and .
Step 11.2.3
Combine the numerators over the common denominator.
Step 11.2.4
Simplify the numerator.
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Step 11.2.4.1
Multiply by .
Step 11.2.4.2
Subtract from .
Step 11.2.5
To write as a fraction with a common denominator, multiply by .
Step 11.2.6
Combine and .
Step 11.2.7
Combine the numerators over the common denominator.
Step 11.2.8
Simplify the numerator.
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Step 11.2.8.1
Multiply by .
Step 11.2.8.2
Subtract from .
Step 11.2.9
Move the negative in front of the fraction.
Step 11.2.10
Use the power rule to distribute the exponent.
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Step 11.2.10.1
Apply the product rule to .
Step 11.2.10.2
Apply the product rule to .
Step 11.2.11
Simplify the expression.
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Step 11.2.11.1
Raise to the power of .
Step 11.2.11.2
Multiply by .
Step 11.2.12
Combine.
Step 11.2.13
Multiply by by adding the exponents.
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Step 11.2.13.1
Multiply by .
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Step 11.2.13.1.1
Raise to the power of .
Step 11.2.13.1.2
Use the power rule to combine exponents.
Step 11.2.13.2
Add and .
Step 11.2.14
Multiply by .
Step 11.2.15
Raise to the power of .
Step 11.2.16
Raise to the power of .
Step 11.2.17
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
Multiply by .
Step 13.2
Subtract from .
Step 14
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 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
Multiply by .
Step 15.2.3
Subtract from .
Step 15.2.4
Raising to any positive power yields .
Step 15.2.5
The final answer is .
Step 16
These are the local extrema for .
is a local maxima
is a local minima
Step 17