Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=( natural log of x)/x
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
The derivative of with respect to is .
Step 1.3
Differentiate using the Power Rule.
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Step 1.3.1
Combine and .
Step 1.3.2
Cancel the common factor of .
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Step 1.3.2.1
Cancel the common factor.
Step 1.3.2.2
Rewrite the expression.
Step 1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.3.4
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
Add and .
Step 2.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
The derivative of with respect to is .
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
Raise to the power of .
Step 2.4.2.2.2
Factor out of .
Step 2.4.2.2.3
Cancel the common factor.
Step 2.4.2.2.4
Rewrite the expression.
Step 2.4.2.2.5
Divide by .
Step 2.4.3
Differentiate using the Power Rule which states that is where .
Step 2.4.4
Simplify with factoring out.
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Step 2.4.4.1
Multiply by .
Step 2.4.4.2
Factor out of .
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Step 2.4.4.2.1
Factor out of .
Step 2.4.4.2.2
Factor out of .
Step 2.4.4.2.3
Factor out of .
Step 2.5
Cancel the common factors.
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Step 2.5.1
Factor out of .
Step 2.5.2
Cancel the common factor.
Step 2.5.3
Rewrite the expression.
Step 2.6
Simplify.
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Step 2.6.1
Apply the distributive property.
Step 2.6.2
Simplify the numerator.
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Step 2.6.2.1
Simplify each term.
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Step 2.6.2.1.1
Multiply by .
Step 2.6.2.1.2
Multiply .
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Step 2.6.2.1.2.1
Multiply by .
Step 2.6.2.1.2.2
Simplify by moving inside the logarithm.
Step 2.6.2.2
Subtract from .
Step 2.6.3
Rewrite as .
Step 2.6.4
Factor out of .
Step 2.6.5
Factor out of .
Step 2.6.6
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
The derivative of with respect to is .
Step 4.1.3
Differentiate using the Power Rule.
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Step 4.1.3.1
Combine and .
Step 4.1.3.2
Cancel the common factor of .
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Step 4.1.3.2.1
Cancel the common factor.
Step 4.1.3.2.2
Rewrite the expression.
Step 4.1.3.3
Differentiate using the Power Rule which states that is where .
Step 4.1.3.4
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
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
Rewrite the equation as .
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 positive real numbers.
Step 6.2.2.3
Plus or minus is .
Step 6.3
Set the argument in less than or equal to to find where the expression is undefined.
Step 6.4
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
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
Simplify the numerator.
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Step 9.1
Use logarithm rules to move out of the exponent.
Step 9.2
The natural logarithm of is .
Step 9.3
Multiply by .
Step 9.4
Multiply by .
Step 9.5
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
The natural logarithm of is .
Step 11.2.2
The final answer is .
Step 12
These are the local extrema for .
is a local maxima
Step 13