Calculus Examples

Find the Absolute Max and Min over the Interval g(x)=(x^2+4)/(10x)
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
By the Sum Rule, the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4
Add and .
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
Differentiate using the Power Rule which states that is where .
Step 1.9
Combine fractions.
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Step 1.9.1
Multiply by .
Step 1.9.2
Multiply by .
Step 1.10
Simplify.
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Step 1.10.1
Apply the distributive property.
Step 1.10.2
Simplify the numerator.
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Step 1.10.2.1
Multiply by .
Step 1.10.2.2
Subtract from .
Step 1.10.3
Simplify the numerator.
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Step 1.10.3.1
Rewrite as .
Step 1.10.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2
Find the second derivative of the function.
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Multiply the exponents in .
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Step 2.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2
Multiply by .
Step 2.4
Differentiate using the Product Rule which states that is where and .
Step 2.5
Differentiate.
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Step 2.5.1
By the Sum Rule, the derivative of with respect to is .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.4
Simplify the expression.
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Step 2.5.4.1
Add and .
Step 2.5.4.2
Multiply by .
Step 2.5.5
By the Sum Rule, the derivative of with respect to is .
Step 2.5.6
Differentiate using the Power Rule which states that is where .
Step 2.5.7
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.8
Simplify by adding terms.
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Step 2.5.8.1
Add and .
Step 2.5.8.2
Multiply by .
Step 2.5.8.3
Add and .
Step 2.5.8.4
Simplify by subtracting numbers.
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Step 2.5.8.4.1
Subtract from .
Step 2.5.8.4.2
Add and .
Step 2.6
Multiply by by adding the exponents.
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Step 2.6.1
Move .
Step 2.6.2
Multiply by .
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Step 2.6.2.1
Raise to the power of .
Step 2.6.2.2
Use the power rule to combine exponents.
Step 2.6.3
Add and .
Step 2.7
Move to the left of .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Combine fractions.
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Step 2.9.1
Multiply by .
Step 2.9.2
Multiply by .
Step 2.10
Simplify.
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Step 2.10.1
Apply the distributive property.
Step 2.10.2
Simplify the numerator.
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Step 2.10.2.1
Simplify each term.
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Step 2.10.2.1.1
Multiply by .
Step 2.10.2.1.2
Expand using the FOIL Method.
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Step 2.10.2.1.2.1
Apply the distributive property.
Step 2.10.2.1.2.2
Apply the distributive property.
Step 2.10.2.1.2.3
Apply the distributive property.
Step 2.10.2.1.3
Simplify and combine like terms.
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Step 2.10.2.1.3.1
Simplify each term.
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Step 2.10.2.1.3.1.1
Multiply by by adding the exponents.
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Step 2.10.2.1.3.1.1.1
Move .
Step 2.10.2.1.3.1.1.2
Multiply by .
Step 2.10.2.1.3.1.2
Multiply by .
Step 2.10.2.1.3.1.3
Multiply by .
Step 2.10.2.1.3.2
Subtract from .
Step 2.10.2.1.3.3
Add and .
Step 2.10.2.1.4
Apply the distributive property.
Step 2.10.2.1.5
Multiply by by adding the exponents.
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Step 2.10.2.1.5.1
Move .
Step 2.10.2.1.5.2
Multiply by .
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Step 2.10.2.1.5.2.1
Raise to the power of .
Step 2.10.2.1.5.2.2
Use the power rule to combine exponents.
Step 2.10.2.1.5.3
Add and .
Step 2.10.2.2
Subtract from .
Step 2.10.2.3
Add and .
Step 2.10.3
Combine terms.
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Step 2.10.3.1
Cancel the common factor of and .
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Step 2.10.3.1.1
Factor out of .
Step 2.10.3.1.2
Cancel the common factors.
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Step 2.10.3.1.2.1
Factor out of .
Step 2.10.3.1.2.2
Cancel the common factor.
Step 2.10.3.1.2.3
Rewrite the expression.
Step 2.10.3.2
Cancel the common factor of and .
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Step 2.10.3.2.1
Factor out of .
Step 2.10.3.2.2
Cancel the common factors.
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Step 2.10.3.2.2.1
Factor out of .
Step 2.10.3.2.2.2
Cancel the common factor.
Step 2.10.3.2.2.3
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
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.4
Add and .
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
Differentiate using the Power Rule which states that is where .
Step 4.1.9
Combine fractions.
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Step 4.1.9.1
Multiply by .
Step 4.1.9.2
Multiply by .
Step 4.1.10
Simplify.
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Step 4.1.10.1
Apply the distributive property.
Step 4.1.10.2
Simplify the numerator.
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Step 4.1.10.2.1
Multiply by .
Step 4.1.10.2.2
Subtract from .
Step 4.1.10.3
Simplify the numerator.
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Step 4.1.10.3.1
Rewrite as .
Step 4.1.10.3.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
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
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.3.2
Set equal to and solve for .
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Step 5.3.2.1
Set equal to .
Step 5.3.2.2
Subtract from both sides of the equation.
Step 5.3.3
Set equal to and solve for .
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Step 5.3.3.1
Set equal to .
Step 5.3.3.2
Add to both sides of the equation.
Step 5.3.4
The final solution is all the values that make true.
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
Divide each term in by and simplify.
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Step 6.2.1.1
Divide each term in by .
Step 6.2.1.2
Simplify the left side.
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Step 6.2.1.2.1
Cancel the common factor of .
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Step 6.2.1.2.1.1
Cancel the common factor.
Step 6.2.1.2.1.2
Divide by .
Step 6.2.1.3
Simplify the right side.
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Step 6.2.1.3.1
Divide by .
Step 6.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.2.3
Simplify .
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Step 6.2.3.1
Rewrite as .
Step 6.2.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 6.2.3.3
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
Simplify the expression.
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Step 9.1.1
Raise to the power of .
Step 9.1.2
Multiply by .
Step 9.2
Cancel the common factor of and .
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Step 9.2.1
Factor out of .
Step 9.2.2
Cancel the common factors.
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Step 9.2.2.1
Factor out of .
Step 9.2.2.2
Cancel the common factor.
Step 9.2.2.3
Rewrite the expression.
Step 9.3
Move the negative in front of the fraction.
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
Simplify the numerator.
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Step 11.2.1.1
Raise to the power of .
Step 11.2.1.2
Add and .
Step 11.2.2
Reduce the expression by cancelling the common factors.
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Step 11.2.2.1
Multiply by .
Step 11.2.2.2
Cancel the common factor of and .
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Step 11.2.2.2.1
Factor out of .
Step 11.2.2.2.2
Cancel the common factors.
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Step 11.2.2.2.2.1
Factor out of .
Step 11.2.2.2.2.2
Cancel the common factor.
Step 11.2.2.2.2.3
Rewrite the expression.
Step 11.2.2.3
Move the negative in front of the fraction.
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
Simplify the expression.
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Step 13.1.1
Raise to the power of .
Step 13.1.2
Multiply by .
Step 13.2
Cancel the common factor of and .
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Step 13.2.1
Factor out of .
Step 13.2.2
Cancel the common factors.
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Step 13.2.2.1
Factor out of .
Step 13.2.2.2
Cancel the common factor.
Step 13.2.2.3
Rewrite the expression.
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
Simplify the numerator.
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Step 15.2.1.1
Raise to the power of .
Step 15.2.1.2
Add and .
Step 15.2.2
Reduce the expression by cancelling the common factors.
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Step 15.2.2.1
Multiply by .
Step 15.2.2.2
Cancel the common factor of and .
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Step 15.2.2.2.1
Factor out of .
Step 15.2.2.2.2
Cancel the common factors.
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Step 15.2.2.2.2.1
Factor out of .
Step 15.2.2.2.2.2
Cancel the common factor.
Step 15.2.2.2.2.3
Rewrite the expression.
Step 15.2.3
The final answer is .
Step 16
These are the local extrema for .
is a local maxima
is a local minima
Step 17