Calculus Examples

Find the Absolute Max and Min over the Interval ( natural log of x)/(x^2)
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
Multiply the exponents in .
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Step 1.2.1
Apply the power rule and multiply exponents, .
Step 1.2.2
Multiply by .
Step 1.3
The derivative of with respect to is .
Step 1.4
Differentiate using the Power Rule.
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Step 1.4.1
Combine and .
Step 1.4.2
Cancel the common factor of and .
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Step 1.4.2.1
Factor out of .
Step 1.4.2.2
Cancel the common factors.
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Step 1.4.2.2.1
Raise to the power of .
Step 1.4.2.2.2
Factor out of .
Step 1.4.2.2.3
Cancel the common factor.
Step 1.4.2.2.4
Rewrite the expression.
Step 1.4.2.2.5
Divide by .
Step 1.4.3
Differentiate using the Power Rule which states that is where .
Step 1.4.4
Simplify with factoring out.
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Step 1.4.4.1
Multiply by .
Step 1.4.4.2
Factor out of .
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Step 1.4.4.2.1
Raise to the power of .
Step 1.4.4.2.2
Factor out of .
Step 1.4.4.2.3
Factor out of .
Step 1.4.4.2.4
Factor out of .
Step 1.5
Cancel the common factors.
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Step 1.5.1
Factor out of .
Step 1.5.2
Cancel the common factor.
Step 1.5.3
Rewrite the expression.
Step 1.6
Simplify by moving inside the logarithm.
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
Add and .
Step 2.2.5
Since is constant with respect to , the derivative of with respect to is .
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
The derivative of with respect to is .
Step 2.3.3
Replace all occurrences of with .
Step 2.4
Differentiate using the Power Rule.
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Step 2.4.1
Combine and .
Step 2.4.2
Cancel the common factor of and .
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Step 2.4.2.1
Factor out of .
Step 2.4.2.2
Cancel the common factors.
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Step 2.4.2.2.1
Multiply by .
Step 2.4.2.2.2
Cancel the common factor.
Step 2.4.2.2.3
Rewrite the expression.
Step 2.4.2.2.4
Divide by .
Step 2.4.3
Differentiate using the Power Rule which states that is where .
Step 2.4.4
Multiply by .
Step 2.5
Raise to the power of .
Step 2.6
Raise to the power of .
Step 2.7
Use the power rule to combine exponents.
Step 2.8
Add and .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Simplify with factoring out.
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Step 2.10.1
Multiply by .
Step 2.10.2
Factor out of .
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Step 2.10.2.1
Factor out of .
Step 2.10.2.2
Factor out of .
Step 2.10.2.3
Factor out of .
Step 2.11
Cancel the common factors.
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Step 2.11.1
Factor out of .
Step 2.11.2
Cancel the common factor.
Step 2.11.3
Rewrite the expression.
Step 2.12
Simplify.
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Step 2.12.1
Apply the distributive property.
Step 2.12.2
Simplify the numerator.
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Step 2.12.2.1
Simplify each term.
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Step 2.12.2.1.1
Multiply by .
Step 2.12.2.1.2
Multiply .
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Step 2.12.2.1.2.1
Multiply by .
Step 2.12.2.1.2.2
Simplify by moving inside the logarithm.
Step 2.12.2.1.3
Multiply the exponents in .
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Step 2.12.2.1.3.1
Apply the power rule and multiply exponents, .
Step 2.12.2.1.3.2
Multiply by .
Step 2.12.2.2
Subtract from .
Step 2.12.3
Rewrite as .
Step 2.12.4
Factor out of .
Step 2.12.5
Factor out of .
Step 2.12.6
Move the negative in front of the fraction.
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 Quotient Rule which states that is where and .
Step 4.1.2
Multiply the exponents in .
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Step 4.1.2.1
Apply the power rule and multiply exponents, .
Step 4.1.2.2
Multiply by .
Step 4.1.3
The derivative of with respect to is .
Step 4.1.4
Differentiate using the Power Rule.
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Step 4.1.4.1
Combine and .
Step 4.1.4.2
Cancel the common factor of and .
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Step 4.1.4.2.1
Factor out of .
Step 4.1.4.2.2
Cancel the common factors.
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Step 4.1.4.2.2.1
Raise to the power of .
Step 4.1.4.2.2.2
Factor out of .
Step 4.1.4.2.2.3
Cancel the common factor.
Step 4.1.4.2.2.4
Rewrite the expression.
Step 4.1.4.2.2.5
Divide by .
Step 4.1.4.3
Differentiate using the Power Rule which states that is where .
Step 4.1.4.4
Simplify with factoring out.
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Step 4.1.4.4.1
Multiply by .
Step 4.1.4.4.2
Factor out of .
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Step 4.1.4.4.2.1
Raise to the power of .
Step 4.1.4.4.2.2
Factor out of .
Step 4.1.4.4.2.3
Factor out of .
Step 4.1.4.4.2.4
Factor out of .
Step 4.1.5
Cancel the common factors.
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Step 4.1.5.1
Factor out of .
Step 4.1.5.2
Cancel the common factor.
Step 4.1.5.3
Rewrite the expression.
Step 4.1.6
Simplify by moving inside the logarithm.
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
Dividing two negative values results in a positive value.
Step 5.3.2.2.2
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
To solve for , rewrite the equation using properties of logarithms.
Step 5.3.4
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 5.3.5
Solve for .
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Step 5.3.5.1
Rewrite the equation as .
Step 5.3.5.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.3.5.3
Simplify.
Step 5.3.5.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 5.3.5.4.1
First, use the positive value of the to find the first solution.
Step 5.3.5.4.2
Next, use the negative value of the to find the second solution.
Step 5.3.5.4.3
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
Set the denominator in equal to to find where the expression is undefined.
Step 6.2
Solve for .
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Step 6.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.2.2
Simplify .
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Step 6.2.2.1
Rewrite as .
Step 6.2.2.2
Pull terms out from under the radical, assuming real numbers.
Step 6.3
Set the argument in less than or equal to to find where the expression is undefined.
Step 6.4
Solve for .
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Step 6.4.1
Take the specified root of both sides of the inequality to eliminate the exponent on the left side.
Step 6.4.2
Simplify the equation.
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Step 6.4.2.1
Simplify the left side.
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Step 6.4.2.1.1
Pull terms out from under the radical.
Step 6.4.2.2
Simplify the right side.
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Step 6.4.2.2.1
Simplify .
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Step 6.4.2.2.1.1
Rewrite as .
Step 6.4.2.2.1.2
Pull terms out from under the radical.
Step 6.4.2.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.4.3
Write as a piecewise.
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Step 6.4.3.1
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
Step 6.4.3.2
In the piece where is non-negative, remove the absolute value.
Step 6.4.3.3
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 6.4.3.4
In the piece where is negative, remove the absolute value and multiply by .
Step 6.4.3.5
Write as a piecewise.
Step 6.4.4
Find the intersection of and .
Step 6.4.5
Solve when .
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Step 6.4.5.1
Divide each term in by and simplify.
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Step 6.4.5.1.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 6.4.5.1.2
Simplify the left side.
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Step 6.4.5.1.2.1
Dividing two negative values results in a positive value.
Step 6.4.5.1.2.2
Divide by .
Step 6.4.5.1.3
Simplify the right side.
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Step 6.4.5.1.3.1
Divide by .
Step 6.4.5.2
Find the intersection of and .
No solution
No solution
Step 6.4.6
Find the union of the solutions.
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 the numerator.
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Step 9.1.1
Rewrite as .
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Step 9.1.1.1
Use to rewrite as .
Step 9.1.1.2
Apply the power rule and multiply exponents, .
Step 9.1.1.3
Combine and .
Step 9.1.1.4
Cancel the common factor of and .
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Step 9.1.1.4.1
Factor out of .
Step 9.1.1.4.2
Cancel the common factors.
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Step 9.1.1.4.2.1
Factor out of .
Step 9.1.1.4.2.2
Cancel the common factor.
Step 9.1.1.4.2.3
Rewrite the expression.
Step 9.1.1.4.2.4
Divide by .
Step 9.1.2
Use logarithm rules to move out of the exponent.
Step 9.1.3
The natural logarithm of is .
Step 9.1.4
Multiply by .
Step 9.1.5
Multiply by .
Step 9.1.6
Subtract from .
Step 9.2
Rewrite as .
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Step 9.2.1
Use to rewrite as .
Step 9.2.2
Apply the power rule and multiply exponents, .
Step 9.2.3
Combine and .
Step 9.2.4
Cancel the common factor of and .
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Step 9.2.4.1
Factor out of .
Step 9.2.4.2
Cancel the common factors.
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Step 9.2.4.2.1
Factor out of .
Step 9.2.4.2.2
Cancel the common factor.
Step 9.2.4.2.3
Rewrite the expression.
Step 9.2.4.2.4
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
Rewrite as .
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Step 11.2.1.1
Use to rewrite as .
Step 11.2.1.2
Apply the power rule and multiply exponents, .
Step 11.2.1.3
Combine and .
Step 11.2.1.4
Cancel the common factor of .
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Step 11.2.1.4.1
Cancel the common factor.
Step 11.2.1.4.2
Rewrite the expression.
Step 11.2.1.5
Simplify.
Step 11.2.2
The final answer is .
Step 12
These are the local extrema for .
is a local maxima
Step 13