Calculus Examples

Find the Local Maxima and Minima y=x/(x^2+25)
Step 1
Write as a function.
Step 2
Find the first derivative of the function.
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Step 2.1
Differentiate using the Quotient Rule which states that is where and .
Step 2.2
Differentiate.
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Step 2.2.1
Differentiate using the Power Rule which states that is where .
Step 2.2.2
Multiply by .
Step 2.2.3
By the Sum Rule, the derivative of with respect to is .
Step 2.2.4
Differentiate using the Power Rule which states that is where .
Step 2.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.6
Simplify the expression.
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Step 2.2.6.1
Add and .
Step 2.2.6.2
Multiply by .
Step 2.3
Raise to the power of .
Step 2.4
Raise to the power of .
Step 2.5
Use the power rule to combine exponents.
Step 2.6
Add and .
Step 2.7
Subtract from .
Step 3
Find the second derivative of the function.
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Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Differentiate.
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Step 3.2.1
Multiply the exponents in .
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Step 3.2.1.1
Apply the power rule and multiply exponents, .
Step 3.2.1.2
Multiply by .
Step 3.2.2
By the Sum Rule, the derivative of with respect to is .
Step 3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.4
Differentiate using the Power Rule which states that is where .
Step 3.2.5
Multiply by .
Step 3.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.7
Add and .
Step 3.3
Differentiate using the chain rule, which states that is where and .
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Step 3.3.1
To apply the Chain Rule, set as .
Step 3.3.2
Differentiate using the Power Rule which states that is where .
Step 3.3.3
Replace all occurrences of with .
Step 3.4
Differentiate.
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Step 3.4.1
Multiply by .
Step 3.4.2
By the Sum Rule, the derivative of with respect to is .
Step 3.4.3
Differentiate using the Power Rule which states that is where .
Step 3.4.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.5
Simplify the expression.
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Step 3.4.5.1
Add and .
Step 3.4.5.2
Move to the left of .
Step 3.4.5.3
Multiply by .
Step 3.5
Simplify.
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Step 3.5.1
Apply the distributive property.
Step 3.5.2
Apply the distributive property.
Step 3.5.3
Simplify the numerator.
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Step 3.5.3.1
Simplify each term.
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Step 3.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 3.5.3.1.2
Rewrite as .
Step 3.5.3.1.3
Expand using the FOIL Method.
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Step 3.5.3.1.3.1
Apply the distributive property.
Step 3.5.3.1.3.2
Apply the distributive property.
Step 3.5.3.1.3.3
Apply the distributive property.
Step 3.5.3.1.4
Simplify and combine like terms.
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Step 3.5.3.1.4.1
Simplify each term.
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Step 3.5.3.1.4.1.1
Multiply by by adding the exponents.
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Step 3.5.3.1.4.1.1.1
Use the power rule to combine exponents.
Step 3.5.3.1.4.1.1.2
Add and .
Step 3.5.3.1.4.1.2
Move to the left of .
Step 3.5.3.1.4.1.3
Multiply by .
Step 3.5.3.1.4.2
Add and .
Step 3.5.3.1.5
Apply the distributive property.
Step 3.5.3.1.6
Simplify.
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Step 3.5.3.1.6.1
Multiply by .
Step 3.5.3.1.6.2
Multiply by .
Step 3.5.3.1.7
Apply the distributive property.
Step 3.5.3.1.8
Simplify.
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Step 3.5.3.1.8.1
Multiply by by adding the exponents.
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Step 3.5.3.1.8.1.1
Move .
Step 3.5.3.1.8.1.2
Multiply by .
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Step 3.5.3.1.8.1.2.1
Raise to the power of .
Step 3.5.3.1.8.1.2.2
Use the power rule to combine exponents.
Step 3.5.3.1.8.1.3
Add and .
Step 3.5.3.1.8.2
Multiply by by adding the exponents.
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Step 3.5.3.1.8.2.1
Move .
Step 3.5.3.1.8.2.2
Multiply by .
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Step 3.5.3.1.8.2.2.1
Raise to the power of .
Step 3.5.3.1.8.2.2.2
Use the power rule to combine exponents.
Step 3.5.3.1.8.2.3
Add and .
Step 3.5.3.1.9
Simplify each term.
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Step 3.5.3.1.9.1
Multiply by .
Step 3.5.3.1.9.2
Multiply by .
Step 3.5.3.1.10
Multiply by by adding the exponents.
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Step 3.5.3.1.10.1
Multiply by .
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Step 3.5.3.1.10.1.1
Raise to the power of .
Step 3.5.3.1.10.1.2
Use the power rule to combine exponents.
Step 3.5.3.1.10.2
Add and .
Step 3.5.3.1.11
Expand using the FOIL Method.
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Step 3.5.3.1.11.1
Apply the distributive property.
Step 3.5.3.1.11.2
Apply the distributive property.
Step 3.5.3.1.11.3
Apply the distributive property.
Step 3.5.3.1.12
Simplify and combine like terms.
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Step 3.5.3.1.12.1
Simplify each term.
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Step 3.5.3.1.12.1.1
Multiply by by adding the exponents.
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Step 3.5.3.1.12.1.1.1
Move .
Step 3.5.3.1.12.1.1.2
Use the power rule to combine exponents.
Step 3.5.3.1.12.1.1.3
Add and .
Step 3.5.3.1.12.1.2
Rewrite using the commutative property of multiplication.
Step 3.5.3.1.12.1.3
Multiply by by adding the exponents.
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Step 3.5.3.1.12.1.3.1
Move .
Step 3.5.3.1.12.1.3.2
Multiply by .
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Step 3.5.3.1.12.1.3.2.1
Raise to the power of .
Step 3.5.3.1.12.1.3.2.2
Use the power rule to combine exponents.
Step 3.5.3.1.12.1.3.3
Add and .
Step 3.5.3.1.12.1.4
Multiply by .
Step 3.5.3.1.12.1.5
Multiply by .
Step 3.5.3.1.12.2
Subtract from .
Step 3.5.3.1.12.3
Add and .
Step 3.5.3.2
Add and .
Step 3.5.3.3
Subtract from .
Step 3.5.4
Simplify the numerator.
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Step 3.5.4.1
Factor out of .
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Step 3.5.4.1.1
Factor out of .
Step 3.5.4.1.2
Factor out of .
Step 3.5.4.1.3
Factor out of .
Step 3.5.4.1.4
Factor out of .
Step 3.5.4.1.5
Factor out of .
Step 3.5.4.2
Rewrite as .
Step 3.5.4.3
Let . Substitute for all occurrences of .
Step 3.5.4.4
Factor using the AC method.
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Step 3.5.4.4.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.5.4.4.2
Write the factored form using these integers.
Step 3.5.4.5
Replace all occurrences of with .
Step 3.5.5
Cancel the common factor of and .
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Step 3.5.5.1
Factor out of .
Step 3.5.5.2
Cancel the common factors.
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Step 3.5.5.2.1
Factor out of .
Step 3.5.5.2.2
Cancel the common factor.
Step 3.5.5.2.3
Rewrite the expression.
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Find the first derivative.
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Step 5.1
Find the first derivative.
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Step 5.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 5.1.2
Differentiate.
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Step 5.1.2.1
Differentiate using the Power Rule which states that is where .
Step 5.1.2.2
Multiply by .
Step 5.1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 5.1.2.4
Differentiate using the Power Rule which states that is where .
Step 5.1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.2.6
Simplify the expression.
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Step 5.1.2.6.1
Add and .
Step 5.1.2.6.2
Multiply by .
Step 5.1.3
Raise to the power of .
Step 5.1.4
Raise to the power of .
Step 5.1.5
Use the power rule to combine exponents.
Step 5.1.6
Add and .
Step 5.1.7
Subtract from .
Step 5.2
The first derivative of with respect to is .
Step 6
Set the first derivative equal to then solve the equation .
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Step 6.1
Set the first derivative equal to .
Step 6.2
Set the numerator equal to zero.
Step 6.3
Solve the equation for .
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Step 6.3.1
Subtract from both sides of the equation.
Step 6.3.2
Divide each term in by and simplify.
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Step 6.3.2.1
Divide each term in by .
Step 6.3.2.2
Simplify the left side.
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Step 6.3.2.2.1
Dividing two negative values results in a positive value.
Step 6.3.2.2.2
Divide by .
Step 6.3.2.3
Simplify the right side.
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Step 6.3.2.3.1
Divide by .
Step 6.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.3.4
Simplify .
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Step 6.3.4.1
Rewrite as .
Step 6.3.4.2
Pull terms out from under the radical, assuming positive real numbers.
Step 6.3.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 6.3.5.1
First, use the positive value of the to find the first solution.
Step 6.3.5.2
Next, use the negative value of the to find the second solution.
Step 6.3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 7
Find the values where the derivative is undefined.
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Step 7.1
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 8
Critical points to evaluate.
Step 9
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 10
Evaluate the second derivative.
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Step 10.1
Multiply by .
Step 10.2
Simplify the denominator.
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Step 10.2.1
Raise to the power of .
Step 10.2.2
Add and .
Step 10.2.3
Raise to the power of .
Step 10.3
Simplify the numerator.
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Step 10.3.1
Raise to the power of .
Step 10.3.2
Subtract from .
Step 10.4
Reduce the expression by cancelling the common factors.
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Step 10.4.1
Multiply by .
Step 10.4.2
Cancel the common factor of and .
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Step 10.4.2.1
Factor out of .
Step 10.4.2.2
Cancel the common factors.
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Step 10.4.2.2.1
Factor out of .
Step 10.4.2.2.2
Cancel the common factor.
Step 10.4.2.2.3
Rewrite the expression.
Step 10.4.3
Move the negative in front of the fraction.
Step 11
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 12
Find the y-value when .
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Step 12.1
Replace the variable with in the expression.
Step 12.2
Simplify the result.
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Step 12.2.1
Simplify the denominator.
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Step 12.2.1.1
Raise to the power of .
Step 12.2.1.2
Add and .
Step 12.2.2
Cancel the common factor of and .
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Step 12.2.2.1
Factor out of .
Step 12.2.2.2
Cancel the common factors.
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Step 12.2.2.2.1
Factor out of .
Step 12.2.2.2.2
Cancel the common factor.
Step 12.2.2.2.3
Rewrite the expression.
Step 12.2.3
The final answer is .
Step 13
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 14
Evaluate the second derivative.
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Step 14.1
Multiply by .
Step 14.2
Simplify the denominator.
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Step 14.2.1
Raise to the power of .
Step 14.2.2
Add and .
Step 14.2.3
Raise to the power of .
Step 14.3
Simplify the numerator.
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Step 14.3.1
Raise to the power of .
Step 14.3.2
Subtract from .
Step 14.4
Reduce the expression by cancelling the common factors.
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Step 14.4.1
Multiply by .
Step 14.4.2
Cancel the common factor of and .
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Step 14.4.2.1
Factor out of .
Step 14.4.2.2
Cancel the common factors.
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Step 14.4.2.2.1
Factor out of .
Step 14.4.2.2.2
Cancel the common factor.
Step 14.4.2.2.3
Rewrite the expression.
Step 15
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 16
Find the y-value when .
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Step 16.1
Replace the variable with in the expression.
Step 16.2
Simplify the result.
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Step 16.2.1
Simplify the denominator.
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Step 16.2.1.1
Raise to the power of .
Step 16.2.1.2
Add and .
Step 16.2.2
Reduce the expression by cancelling the common factors.
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Step 16.2.2.1
Cancel the common factor of and .
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Step 16.2.2.1.1
Factor out of .
Step 16.2.2.1.2
Cancel the common factors.
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Step 16.2.2.1.2.1
Factor out of .
Step 16.2.2.1.2.2
Cancel the common factor.
Step 16.2.2.1.2.3
Rewrite the expression.
Step 16.2.2.2
Move the negative in front of the fraction.
Step 16.2.3
The final answer is .
Step 17
These are the local extrema for .
is a local maxima
is a local minima
Step 18