Calculus Examples

Find the Local Maxima and Minima f(x)=xe^(-x^2)
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the Product Rule which states that is where and .
Step 1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.1
To apply the Chain Rule, set as .
Step 1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.2.3
Replace all occurrences of with .
Step 1.3
Differentiate.
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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
Raise to the power of .
Step 1.5
Raise to the power of .
Step 1.6
Use the power rule to combine exponents.
Step 1.7
Simplify the expression.
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Step 1.7.1
Add and .
Step 1.7.2
Move to the left of .
Step 1.8
Differentiate using the Power Rule which states that is where .
Step 1.9
Multiply by .
Step 1.10
Simplify.
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Step 1.10.1
Reorder terms.
Step 1.10.2
Reorder factors in .
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 Product Rule which states that is where and .
Step 2.2.3
Differentiate using the chain rule, which states that is where and .
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Step 2.2.3.1
To apply the Chain Rule, set as .
Step 2.2.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.3.3
Replace all occurrences of with .
Step 2.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.5
Differentiate using the Power Rule which states that is where .
Step 2.2.6
Differentiate using the Power Rule which states that is where .
Step 2.2.7
Multiply by .
Step 2.2.8
Multiply by by adding the exponents.
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Step 2.2.8.1
Move .
Step 2.2.8.2
Multiply by .
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Step 2.2.8.2.1
Raise to the power of .
Step 2.2.8.2.2
Use the power rule to combine exponents.
Step 2.2.8.3
Add and .
Step 2.2.9
Move to the left of .
Step 2.3
Evaluate .
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Step 2.3.1
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1.1
To apply the Chain Rule, set as .
Step 2.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.1.3
Replace all occurrences of with .
Step 2.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.3
Differentiate using the Power Rule which states that is where .
Step 2.3.4
Multiply by .
Step 2.4
Simplify.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Combine terms.
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Step 2.4.2.1
Multiply by .
Step 2.4.2.2
Multiply by .
Step 2.4.2.3
Move .
Step 2.4.2.4
Subtract from .
Step 2.4.3
Reorder terms.
Step 2.4.4
Reorder factors in .
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 Product Rule which states that is where and .
Step 4.1.2
Differentiate using the chain rule, which states that is where and .
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Step 4.1.2.1
To apply the Chain Rule, set as .
Step 4.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.1.2.3
Replace all occurrences of with .
Step 4.1.3
Differentiate.
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Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Multiply by .
Step 4.1.4
Raise to the power of .
Step 4.1.5
Raise to the power of .
Step 4.1.6
Use the power rule to combine exponents.
Step 4.1.7
Simplify the expression.
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Step 4.1.7.1
Add and .
Step 4.1.7.2
Move to the left of .
Step 4.1.8
Differentiate using the Power Rule which states that is where .
Step 4.1.9
Multiply by .
Step 4.1.10
Simplify.
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Step 4.1.10.1
Reorder terms.
Step 4.1.10.2
Reorder factors in .
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 out of .
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Step 5.2.1
Factor out of .
Step 5.2.2
Multiply by .
Step 5.2.3
Factor out of .
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
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 5.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 5.4.2.3
There is no solution for
No solution
No solution
No solution
Step 5.5
Set equal to and solve for .
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Step 5.5.1
Set equal to .
Step 5.5.2
Solve for .
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Step 5.5.2.1
Subtract from both sides of the equation.
Step 5.5.2.2
Divide each term in by and simplify.
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Step 5.5.2.2.1
Divide each term in by .
Step 5.5.2.2.2
Simplify the left side.
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Step 5.5.2.2.2.1
Cancel the common factor of .
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Step 5.5.2.2.2.1.1
Cancel the common factor.
Step 5.5.2.2.2.1.2
Divide by .
Step 5.5.2.2.3
Simplify the right side.
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Step 5.5.2.2.3.1
Dividing two negative values results in a positive value.
Step 5.5.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.5.2.4
Simplify .
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Step 5.5.2.4.1
Rewrite as .
Step 5.5.2.4.2
Any root of is .
Step 5.5.2.4.3
Multiply by .
Step 5.5.2.4.4
Combine and simplify the denominator.
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Step 5.5.2.4.4.1
Multiply by .
Step 5.5.2.4.4.2
Raise to the power of .
Step 5.5.2.4.4.3
Raise to the power of .
Step 5.5.2.4.4.4
Use the power rule to combine exponents.
Step 5.5.2.4.4.5
Add and .
Step 5.5.2.4.4.6
Rewrite as .
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Step 5.5.2.4.4.6.1
Use to rewrite as .
Step 5.5.2.4.4.6.2
Apply the power rule and multiply exponents, .
Step 5.5.2.4.4.6.3
Combine and .
Step 5.5.2.4.4.6.4
Cancel the common factor of .
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Step 5.5.2.4.4.6.4.1
Cancel the common factor.
Step 5.5.2.4.4.6.4.2
Rewrite the expression.
Step 5.5.2.4.4.6.5
Evaluate the exponent.
Step 5.5.2.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 5.5.2.5.1
First, use the positive value of the to find the first solution.
Step 5.5.2.5.2
Next, use the negative value of the to find the second solution.
Step 5.5.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Apply the product rule to .
Step 9.1.2
Simplify the numerator.
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Step 9.1.2.1
Rewrite as .
Step 9.1.2.2
Raise to the power of .
Step 9.1.2.3
Rewrite as .
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Step 9.1.2.3.1
Factor out of .
Step 9.1.2.3.2
Rewrite as .
Step 9.1.2.4
Pull terms out from under the radical.
Step 9.1.3
Raise to the power of .
Step 9.1.4
Cancel the common factor of .
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Step 9.1.4.1
Factor out of .
Step 9.1.4.2
Cancel the common factor.
Step 9.1.4.3
Rewrite the expression.
Step 9.1.5
Cancel the common factor of .
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Step 9.1.5.1
Cancel the common factor.
Step 9.1.5.2
Divide by .
Step 9.1.6
Apply the product rule to .
Step 9.1.7
Rewrite as .
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Step 9.1.7.1
Use to rewrite as .
Step 9.1.7.2
Apply the power rule and multiply exponents, .
Step 9.1.7.3
Combine and .
Step 9.1.7.4
Cancel the common factor of .
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Step 9.1.7.4.1
Cancel the common factor.
Step 9.1.7.4.2
Rewrite the expression.
Step 9.1.7.5
Evaluate the exponent.
Step 9.1.8
Raise to the power of .
Step 9.1.9
Cancel the common factor of and .
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Step 9.1.9.1
Factor out of .
Step 9.1.9.2
Cancel the common factors.
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Step 9.1.9.2.1
Factor out of .
Step 9.1.9.2.2
Cancel the common factor.
Step 9.1.9.2.3
Rewrite the expression.
Step 9.1.10
Rewrite the expression using the negative exponent rule .
Step 9.1.11
Combine and .
Step 9.1.12
Cancel the common factor of .
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Step 9.1.12.1
Factor out of .
Step 9.1.12.2
Cancel the common factor.
Step 9.1.12.3
Rewrite the expression.
Step 9.1.13
Apply the product rule to .
Step 9.1.14
Rewrite as .
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Step 9.1.14.1
Use to rewrite as .
Step 9.1.14.2
Apply the power rule and multiply exponents, .
Step 9.1.14.3
Combine and .
Step 9.1.14.4
Cancel the common factor of .
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Step 9.1.14.4.1
Cancel the common factor.
Step 9.1.14.4.2
Rewrite the expression.
Step 9.1.14.5
Evaluate the exponent.
Step 9.1.15
Raise to the power of .
Step 9.1.16
Cancel the common factor of and .
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Step 9.1.16.1
Factor out of .
Step 9.1.16.2
Cancel the common factors.
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Step 9.1.16.2.1
Factor out of .
Step 9.1.16.2.2
Cancel the common factor.
Step 9.1.16.2.3
Rewrite the expression.
Step 9.1.17
Rewrite the expression using the negative exponent rule .
Step 9.1.18
Multiply .
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Step 9.1.18.1
Combine and .
Step 9.1.18.2
Combine and .
Step 9.1.19
Move the negative in front of the fraction.
Step 9.2
Simplify terms.
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Step 9.2.1
Combine the numerators over the common denominator.
Step 9.2.2
Subtract from .
Step 9.2.3
Move the negative in front of the fraction.
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
Apply the product rule to .
Step 11.2.2
Rewrite as .
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Step 11.2.2.1
Use to rewrite as .
Step 11.2.2.2
Apply the power rule and multiply exponents, .
Step 11.2.2.3
Combine and .
Step 11.2.2.4
Cancel the common factor of .
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Step 11.2.2.4.1
Cancel the common factor.
Step 11.2.2.4.2
Rewrite the expression.
Step 11.2.2.5
Evaluate the exponent.
Step 11.2.3
Raise to the power of .
Step 11.2.4
Cancel the common factor of and .
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Step 11.2.4.1
Factor out of .
Step 11.2.4.2
Cancel the common factors.
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Step 11.2.4.2.1
Factor out of .
Step 11.2.4.2.2
Cancel the common factor.
Step 11.2.4.2.3
Rewrite the expression.
Step 11.2.5
Rewrite the expression using the negative exponent rule .
Step 11.2.6
Combine.
Step 11.2.7
Multiply by .
Step 11.2.8
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
Use the power rule to distribute the exponent.
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Step 13.1.1.1
Apply the product rule to .
Step 13.1.1.2
Apply the product rule to .
Step 13.1.2
Raise to the power of .
Step 13.1.3
Simplify the numerator.
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Step 13.1.3.1
Rewrite as .
Step 13.1.3.2
Raise to the power of .
Step 13.1.3.3
Rewrite as .
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Step 13.1.3.3.1
Factor out of .
Step 13.1.3.3.2
Rewrite as .
Step 13.1.3.4
Pull terms out from under the radical.
Step 13.1.4
Raise to the power of .
Step 13.1.5
Cancel the common factor of .
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Step 13.1.5.1
Move the leading negative in into the numerator.
Step 13.1.5.2
Factor out of .
Step 13.1.5.3
Cancel the common factor.
Step 13.1.5.4
Rewrite the expression.
Step 13.1.6
Cancel the common factor of and .
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Step 13.1.6.1
Factor out of .
Step 13.1.6.2
Cancel the common factors.
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Step 13.1.6.2.1
Factor out of .
Step 13.1.6.2.2
Cancel the common factor.
Step 13.1.6.2.3
Rewrite the expression.
Step 13.1.6.2.4
Divide by .
Step 13.1.7
Use the power rule to distribute the exponent.
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Step 13.1.7.1
Apply the product rule to .
Step 13.1.7.2
Apply the product rule to .
Step 13.1.8
Multiply by by adding the exponents.
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Step 13.1.8.1
Move .
Step 13.1.8.2
Multiply by .
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Step 13.1.8.2.1
Raise to the power of .
Step 13.1.8.2.2
Use the power rule to combine exponents.
Step 13.1.8.3
Add and .
Step 13.1.9
Raise to the power of .
Step 13.1.10
Rewrite as .
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Step 13.1.10.1
Use to rewrite as .
Step 13.1.10.2
Apply the power rule and multiply exponents, .
Step 13.1.10.3
Combine and .
Step 13.1.10.4
Cancel the common factor of .
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Step 13.1.10.4.1
Cancel the common factor.
Step 13.1.10.4.2
Rewrite the expression.
Step 13.1.10.5
Evaluate the exponent.
Step 13.1.11
Raise to the power of .
Step 13.1.12
Cancel the common factor of and .
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Step 13.1.12.1
Factor out of .
Step 13.1.12.2
Cancel the common factors.
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Step 13.1.12.2.1
Factor out of .
Step 13.1.12.2.2
Cancel the common factor.
Step 13.1.12.2.3
Rewrite the expression.
Step 13.1.13
Rewrite the expression using the negative exponent rule .
Step 13.1.14
Combine and .
Step 13.1.15
Cancel the common factor of .
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Step 13.1.15.1
Move the leading negative in into the numerator.
Step 13.1.15.2
Factor out of .
Step 13.1.15.3
Cancel the common factor.
Step 13.1.15.4
Rewrite the expression.
Step 13.1.16
Multiply by .
Step 13.1.17
Use the power rule to distribute the exponent.
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Step 13.1.17.1
Apply the product rule to .
Step 13.1.17.2
Apply the product rule to .
Step 13.1.18
Multiply by by adding the exponents.
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Step 13.1.18.1
Move .
Step 13.1.18.2
Multiply by .
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Step 13.1.18.2.1
Raise to the power of .
Step 13.1.18.2.2
Use the power rule to combine exponents.
Step 13.1.18.3
Add and .
Step 13.1.19
Raise to the power of .
Step 13.1.20
Rewrite as .
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Step 13.1.20.1
Use to rewrite as .
Step 13.1.20.2
Apply the power rule and multiply exponents, .
Step 13.1.20.3
Combine and .
Step 13.1.20.4
Cancel the common factor of .
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Step 13.1.20.4.1
Cancel the common factor.
Step 13.1.20.4.2
Rewrite the expression.
Step 13.1.20.5
Evaluate the exponent.
Step 13.1.21
Raise to the power of .
Step 13.1.22
Cancel the common factor of and .
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Step 13.1.22.1
Factor out of .
Step 13.1.22.2
Cancel the common factors.
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Step 13.1.22.2.1
Factor out of .
Step 13.1.22.2.2
Cancel the common factor.
Step 13.1.22.2.3
Rewrite the expression.
Step 13.1.23
Rewrite the expression using the negative exponent rule .
Step 13.1.24
Multiply .
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Step 13.1.24.1
Combine and .
Step 13.1.24.2
Combine and .
Step 13.2
Simplify terms.
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Step 13.2.1
Combine the numerators over the common denominator.
Step 13.2.2
Add and .
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
Use the power rule to distribute the exponent.
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Step 15.2.1.1
Apply the product rule to .
Step 15.2.1.2
Apply the product rule to .
Step 15.2.2
Multiply by by adding the exponents.
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Step 15.2.2.1
Move .
Step 15.2.2.2
Multiply by .
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Step 15.2.2.2.1
Raise to the power of .
Step 15.2.2.2.2
Use the power rule to combine exponents.
Step 15.2.2.3
Add and .
Step 15.2.3
Raise to the power of .
Step 15.2.4
Rewrite as .
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Step 15.2.4.1
Use to rewrite as .
Step 15.2.4.2
Apply the power rule and multiply exponents, .
Step 15.2.4.3
Combine and .
Step 15.2.4.4
Cancel the common factor of .
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Step 15.2.4.4.1
Cancel the common factor.
Step 15.2.4.4.2
Rewrite the expression.
Step 15.2.4.5
Evaluate the exponent.
Step 15.2.5
Raise to the power of .
Step 15.2.6
Cancel the common factor of and .
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Step 15.2.6.1
Factor out of .
Step 15.2.6.2
Cancel the common factors.
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Step 15.2.6.2.1
Factor out of .
Step 15.2.6.2.2
Cancel the common factor.
Step 15.2.6.2.3
Rewrite the expression.
Step 15.2.7
Rewrite the expression using the negative exponent rule .
Step 15.2.8
Multiply by .
Step 15.2.9
Move to the left of .
Step 15.2.10
The final answer is .
Step 16
These are the local extrema for .
is a local maxima
is a local minima
Step 17