Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=(4x)/(x^2+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 Quotient Rule which states that is where and .
Step 1.3
Differentiate.
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Step 1.3.1
Differentiate using the Power Rule which states that is where .
Step 1.3.2
Multiply by .
Step 1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.6
Simplify the expression.
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Step 1.3.6.1
Add and .
Step 1.3.6.2
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
Add and .
Step 1.8
Subtract from .
Step 1.9
Combine and .
Step 1.10
Simplify.
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Step 1.10.1
Apply the distributive property.
Step 1.10.2
Simplify each term.
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Step 1.10.2.1
Multiply by .
Step 1.10.2.2
Multiply by .
Step 2
Find the second 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
Multiply the exponents in .
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Step 2.2.1.1
Apply the power rule and multiply exponents, .
Step 2.2.1.2
Multiply by .
Step 2.2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.2.3
Since is constant with respect to , 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
Multiply by .
Step 2.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.7
Add and .
Step 2.3
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1
To apply the Chain Rule, set as .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Differentiate.
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Step 2.4.1
Multiply by .
Step 2.4.2
By the Sum Rule, the derivative of with respect to is .
Step 2.4.3
Differentiate using the Power Rule which states that is where .
Step 2.4.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.5
Simplify the expression.
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Step 2.4.5.1
Add and .
Step 2.4.5.2
Move to the left of .
Step 2.4.5.3
Multiply by .
Step 2.5
Simplify.
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Step 2.5.1
Apply the distributive property.
Step 2.5.2
Apply the distributive property.
Step 2.5.3
Simplify the numerator.
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Step 2.5.3.1
Simplify each term.
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Step 2.5.3.1.1
Rewrite using the commutative property of multiplication.
Step 2.5.3.1.2
Rewrite as .
Step 2.5.3.1.3
Expand using the FOIL Method.
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Step 2.5.3.1.3.1
Apply the distributive property.
Step 2.5.3.1.3.2
Apply the distributive property.
Step 2.5.3.1.3.3
Apply the distributive property.
Step 2.5.3.1.4
Simplify and combine like terms.
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Step 2.5.3.1.4.1
Simplify each term.
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Step 2.5.3.1.4.1.1
Multiply by by adding the exponents.
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Step 2.5.3.1.4.1.1.1
Use the power rule to combine exponents.
Step 2.5.3.1.4.1.1.2
Add and .
Step 2.5.3.1.4.1.2
Multiply by .
Step 2.5.3.1.4.1.3
Multiply by .
Step 2.5.3.1.4.1.4
Multiply by .
Step 2.5.3.1.4.2
Add and .
Step 2.5.3.1.5
Apply the distributive property.
Step 2.5.3.1.6
Simplify.
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Step 2.5.3.1.6.1
Multiply by .
Step 2.5.3.1.6.2
Multiply by .
Step 2.5.3.1.7
Apply the distributive property.
Step 2.5.3.1.8
Simplify.
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Step 2.5.3.1.8.1
Multiply by by adding the exponents.
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Step 2.5.3.1.8.1.1
Move .
Step 2.5.3.1.8.1.2
Multiply by .
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Step 2.5.3.1.8.1.2.1
Raise to the power of .
Step 2.5.3.1.8.1.2.2
Use the power rule to combine exponents.
Step 2.5.3.1.8.1.3
Add and .
Step 2.5.3.1.8.2
Multiply by by adding the exponents.
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Step 2.5.3.1.8.2.1
Move .
Step 2.5.3.1.8.2.2
Multiply by .
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Step 2.5.3.1.8.2.2.1
Raise to the power of .
Step 2.5.3.1.8.2.2.2
Use the power rule to combine exponents.
Step 2.5.3.1.8.2.3
Add and .
Step 2.5.3.1.9
Simplify each term.
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Step 2.5.3.1.9.1
Multiply by .
Step 2.5.3.1.9.2
Multiply by .
Step 2.5.3.1.10
Simplify each term.
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Step 2.5.3.1.10.1
Multiply by by adding the exponents.
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Step 2.5.3.1.10.1.1
Multiply by .
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Step 2.5.3.1.10.1.1.1
Raise to the power of .
Step 2.5.3.1.10.1.1.2
Use the power rule to combine exponents.
Step 2.5.3.1.10.1.2
Add and .
Step 2.5.3.1.10.2
Multiply by .
Step 2.5.3.1.11
Expand using the FOIL Method.
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Step 2.5.3.1.11.1
Apply the distributive property.
Step 2.5.3.1.11.2
Apply the distributive property.
Step 2.5.3.1.11.3
Apply the distributive property.
Step 2.5.3.1.12
Simplify and combine like terms.
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Step 2.5.3.1.12.1
Simplify each term.
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Step 2.5.3.1.12.1.1
Multiply by by adding the exponents.
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Step 2.5.3.1.12.1.1.1
Move .
Step 2.5.3.1.12.1.1.2
Use the power rule to combine exponents.
Step 2.5.3.1.12.1.1.3
Add and .
Step 2.5.3.1.12.1.2
Multiply by by adding the exponents.
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Step 2.5.3.1.12.1.2.1
Move .
Step 2.5.3.1.12.1.2.2
Multiply by .
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Step 2.5.3.1.12.1.2.2.1
Raise to the power of .
Step 2.5.3.1.12.1.2.2.2
Use the power rule to combine exponents.
Step 2.5.3.1.12.1.2.3
Add and .
Step 2.5.3.1.12.2
Subtract from .
Step 2.5.3.1.12.3
Add and .
Step 2.5.3.2
Add and .
Step 2.5.3.3
Subtract from .
Step 2.5.4
Simplify the numerator.
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Step 2.5.4.1
Factor out of .
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Step 2.5.4.1.1
Factor out of .
Step 2.5.4.1.2
Factor out of .
Step 2.5.4.1.3
Factor out of .
Step 2.5.4.1.4
Factor out of .
Step 2.5.4.1.5
Factor out of .
Step 2.5.4.2
Rewrite as .
Step 2.5.4.3
Let . Substitute for all occurrences of .
Step 2.5.4.4
Factor using the AC method.
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Step 2.5.4.4.1
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 2.5.4.4.2
Write the factored form using these integers.
Step 2.5.4.5
Replace all occurrences of with .
Step 2.5.5
Cancel the common factor of and .
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Step 2.5.5.1
Factor out of .
Step 2.5.5.2
Cancel the common factors.
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Step 2.5.5.2.1
Factor out of .
Step 2.5.5.2.2
Cancel the common factor.
Step 2.5.5.2.3
Rewrite the expression.
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 Quotient Rule which states that is where and .
Step 4.1.3
Differentiate.
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Step 4.1.3.1
Differentiate using the Power Rule which states that is where .
Step 4.1.3.2
Multiply by .
Step 4.1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3.4
Differentiate using the Power Rule which states that is where .
Step 4.1.3.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.6
Simplify the expression.
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Step 4.1.3.6.1
Add and .
Step 4.1.3.6.2
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
Add and .
Step 4.1.8
Subtract from .
Step 4.1.9
Combine and .
Step 4.1.10
Simplify.
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Step 4.1.10.1
Apply the distributive property.
Step 4.1.10.2
Simplify each term.
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Step 4.1.10.2.1
Multiply by .
Step 4.1.10.2.2
Multiply by .
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
Subtract from both sides of the equation.
Step 5.3.2
Divide each term in by and simplify.
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Step 5.3.2.1
Divide each term in by .
Step 5.3.2.2
Simplify the left side.
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Step 5.3.2.2.1
Cancel the common factor of .
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Step 5.3.2.2.1.1
Cancel the common factor.
Step 5.3.2.2.1.2
Divide by .
Step 5.3.2.3
Simplify the right side.
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Step 5.3.2.3.1
Divide by .
Step 5.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.3.4
Any root of is .
Step 5.3.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 5.3.5.1
First, use the positive value of the to find the first solution.
Step 5.3.5.2
Next, use the negative value of the to find the second solution.
Step 5.3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Multiply by .
Step 9.2
Simplify the denominator.
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Step 9.2.1
One to any power is one.
Step 9.2.2
Add and .
Step 9.2.3
Raise to the power of .
Step 9.3
Simplify the numerator.
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Step 9.3.1
One to any power is one.
Step 9.3.2
Subtract from .
Step 9.4
Simplify the expression.
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Step 9.4.1
Multiply by .
Step 9.4.2
Divide by .
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
Multiply by .
Step 11.2.2
Simplify the denominator.
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Step 11.2.2.1
One to any power is one.
Step 11.2.2.2
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
Multiply by .
Step 13.2
Simplify the denominator.
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Step 13.2.1
Raise to the power of .
Step 13.2.2
Add and .
Step 13.2.3
Raise to the power of .
Step 13.3
Simplify the numerator.
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Step 13.3.1
Raise to the power of .
Step 13.3.2
Subtract from .
Step 13.4
Simplify the expression.
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Step 13.4.1
Multiply by .
Step 13.4.2
Divide by .
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
Multiply by .
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
Add and .
Step 15.2.3
Divide by .
Step 15.2.4
The final answer is .
Step 16
These are the local extrema for .
is a local maxima
is a local minima
Step 17