Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x+1/x
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate.
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Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.2
Evaluate .
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Step 1.2.1
Rewrite as .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.3
Rewrite the expression using the negative exponent rule .
Step 1.4
Reorder terms.
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
Differentiate using the Product Rule which states that is where and .
Step 2.2.2
Rewrite as .
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 Power Rule which states that is where .
Step 2.2.3.3
Replace all occurrences of with .
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
Multiply the exponents in .
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Step 2.2.6.1
Apply the power rule and multiply exponents, .
Step 2.2.6.2
Multiply by .
Step 2.2.7
Multiply by .
Step 2.2.8
Raise to the power of .
Step 2.2.9
Use the power rule to combine exponents.
Step 2.2.10
Subtract from .
Step 2.2.11
Multiply by .
Step 2.2.12
Multiply by .
Step 2.2.13
Add and .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Simplify.
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Step 2.4.1
Rewrite the expression using the negative exponent rule .
Step 2.4.2
Combine terms.
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Step 2.4.2.1
Combine and .
Step 2.4.2.2
Add and .
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.
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Step 4.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.1.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2
Evaluate .
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Step 4.1.2.1
Rewrite as .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3
Rewrite the expression using the negative exponent rule .
Step 4.1.4
Reorder terms.
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
Subtract from both sides of the equation.
Step 5.3
Find the LCD of the terms in the equation.
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Step 5.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 5.3.2
The LCM of one and any expression is the expression.
Step 5.4
Multiply each term in by to eliminate the fractions.
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Step 5.4.1
Multiply each term in by .
Step 5.4.2
Simplify the left side.
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Step 5.4.2.1
Cancel the common factor of .
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Step 5.4.2.1.1
Move the leading negative in into the numerator.
Step 5.4.2.1.2
Cancel the common factor.
Step 5.4.2.1.3
Rewrite the expression.
Step 5.5
Solve the equation.
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Step 5.5.1
Rewrite the equation as .
Step 5.5.2
Divide each term in by and simplify.
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Step 5.5.2.1
Divide each term in by .
Step 5.5.2.2
Simplify the left side.
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Step 5.5.2.2.1
Dividing two negative values results in a positive value.
Step 5.5.2.2.2
Divide by .
Step 5.5.2.3
Simplify the right side.
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Step 5.5.2.3.1
Divide by .
Step 5.5.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.5.4
Any root of is .
Step 5.5.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 5.5.5.1
First, use the positive value of the to find the first solution.
Step 5.5.5.2
Next, use the negative value of the to find the second solution.
Step 5.5.5.3
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
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 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
One to any power is one.
Step 9.2
Divide by .
Step 10
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 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
Divide by .
Step 11.2.2
Add and .
Step 11.2.3
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
Raise to the power of .
Step 13.2
Divide by .
Step 14
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 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
Divide by .
Step 15.2.2
Subtract from .
Step 15.2.3
The final answer is .
Step 16
These are the local extrema for .
is a local minima
is a local maxima
Step 17