Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=(2x^(5/2))/5-(4x^(3/2))/3+(x^2)/2-5 , [0,4]
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Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
Evaluate .
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Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.2.4
Combine and .
Step 1.1.1.2.5
Combine the numerators over the common denominator.
Step 1.1.1.2.6
Simplify the numerator.
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Step 1.1.1.2.6.1
Multiply by .
Step 1.1.1.2.6.2
Subtract from .
Step 1.1.1.2.7
Combine and .
Step 1.1.1.2.8
Multiply by .
Step 1.1.1.2.9
Multiply by .
Step 1.1.1.2.10
Multiply by .
Step 1.1.1.2.11
Cancel the common factor.
Step 1.1.1.2.12
Divide by .
Step 1.1.1.3
Evaluate .
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Step 1.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.3.4
Combine and .
Step 1.1.1.3.5
Combine the numerators over the common denominator.
Step 1.1.1.3.6
Simplify the numerator.
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Step 1.1.1.3.6.1
Multiply by .
Step 1.1.1.3.6.2
Subtract from .
Step 1.1.1.3.7
Combine and .
Step 1.1.1.3.8
Multiply by .
Step 1.1.1.3.9
Multiply by .
Step 1.1.1.3.10
Multiply by .
Step 1.1.1.3.11
Factor out of .
Step 1.1.1.3.12
Cancel the common factors.
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Step 1.1.1.3.12.1
Factor out of .
Step 1.1.1.3.12.2
Cancel the common factor.
Step 1.1.1.3.12.3
Rewrite the expression.
Step 1.1.1.3.12.4
Divide by .
Step 1.1.1.3.13
Multiply by .
Step 1.1.1.4
Evaluate .
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Step 1.1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.4.3
Combine and .
Step 1.1.1.4.4
Combine and .
Step 1.1.1.4.5
Cancel the common factor of .
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Step 1.1.1.4.5.1
Cancel the common factor.
Step 1.1.1.4.5.2
Divide by .
Step 1.1.1.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.6
Simplify.
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Step 1.1.1.6.1
Add and .
Step 1.1.1.6.2
Reorder terms.
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Find a common factor that is present in each term.
Step 1.2.3
Substitute for .
Step 1.2.4
Solve for .
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Step 1.2.4.1
Multiply by by adding the exponents.
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Step 1.2.4.1.1
Use the power rule to combine exponents.
Step 1.2.4.1.2
Add and .
Step 1.2.4.2
Factor out of .
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Step 1.2.4.2.1
Factor out of .
Step 1.2.4.2.2
Factor out of .
Step 1.2.4.2.3
Factor out of .
Step 1.2.4.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4.4
Set equal to .
Step 1.2.4.5
Set equal to and solve for .
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Step 1.2.4.5.1
Set equal to .
Step 1.2.4.5.2
Solve for .
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Step 1.2.4.5.2.1
Add to both sides of the equation.
Step 1.2.4.5.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.4.5.2.3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.4.5.2.3.1
First, use the positive value of the to find the first solution.
Step 1.2.4.5.2.3.2
Next, use the negative value of the to find the second solution.
Step 1.2.4.5.2.3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.2.4.6
The final solution is all the values that make true.
Step 1.2.5
Substitute for .
Step 1.2.6
Solve for for .
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Step 1.2.6.1
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 1.2.6.2
Simplify the exponent.
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Step 1.2.6.2.1
Simplify the left side.
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Step 1.2.6.2.1.1
Simplify .
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Step 1.2.6.2.1.1.1
Multiply the exponents in .
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Step 1.2.6.2.1.1.1.1
Apply the power rule and multiply exponents, .
Step 1.2.6.2.1.1.1.2
Cancel the common factor of .
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Step 1.2.6.2.1.1.1.2.1
Cancel the common factor.
Step 1.2.6.2.1.1.1.2.2
Rewrite the expression.
Step 1.2.6.2.1.1.2
Simplify.
Step 1.2.6.2.2
Simplify the right side.
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Step 1.2.6.2.2.1
Raising to any positive power yields .
Step 1.2.7
Solve for for .
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Step 1.2.7.1
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 1.2.7.2
Simplify the exponent.
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Step 1.2.7.2.1
Simplify the left side.
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Step 1.2.7.2.1.1
Simplify .
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Step 1.2.7.2.1.1.1
Multiply the exponents in .
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Step 1.2.7.2.1.1.1.1
Apply the power rule and multiply exponents, .
Step 1.2.7.2.1.1.1.2
Cancel the common factor of .
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Step 1.2.7.2.1.1.1.2.1
Cancel the common factor.
Step 1.2.7.2.1.1.1.2.2
Rewrite the expression.
Step 1.2.7.2.1.1.2
Simplify.
Step 1.2.7.2.2
Simplify the right side.
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Step 1.2.7.2.2.1
Rewrite as .
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Step 1.2.7.2.2.1.1
Use to rewrite as .
Step 1.2.7.2.2.1.2
Apply the power rule and multiply exponents, .
Step 1.2.7.2.2.1.3
Combine and .
Step 1.2.7.2.2.1.4
Cancel the common factor of and .
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Step 1.2.7.2.2.1.4.1
Factor out of .
Step 1.2.7.2.2.1.4.2
Cancel the common factors.
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Step 1.2.7.2.2.1.4.2.1
Factor out of .
Step 1.2.7.2.2.1.4.2.2
Cancel the common factor.
Step 1.2.7.2.2.1.4.2.3
Rewrite the expression.
Step 1.2.7.2.2.1.5
Rewrite as .
Step 1.2.8
List all of the solutions.
Step 1.2.9
Exclude the solutions that do not make true.
Step 1.3
Find the values where the derivative is undefined.
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Step 1.3.1
Convert expressions with fractional exponents to radicals.
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Step 1.3.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 1.3.1.2
Apply the rule to rewrite the exponentiation as a radical.
Step 1.3.1.3
Anything raised to is the base itself.
Step 1.3.2
Set the radicand in less than to find where the expression is undefined.
Step 1.3.3
Solve for .
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Step 1.3.3.1
Take the specified root of both sides of the inequality to eliminate the exponent on the left side.
Step 1.3.3.2
Simplify the equation.
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Step 1.3.3.2.1
Simplify the left side.
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Step 1.3.3.2.1.1
Pull terms out from under the radical.
Step 1.3.3.2.2
Simplify the right side.
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Step 1.3.3.2.2.1
Simplify .
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Step 1.3.3.2.2.1.1
Rewrite as .
Step 1.3.3.2.2.1.2
Pull terms out from under the radical.
Step 1.3.4
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 1.4
Evaluate at each value where the derivative is or undefined.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
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Step 1.4.1.2.1
Simplify each term.
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Step 1.4.1.2.1.1
Simplify the numerator.
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Step 1.4.1.2.1.1.1
Rewrite as .
Step 1.4.1.2.1.1.2
Apply the power rule and multiply exponents, .
Step 1.4.1.2.1.1.3
Cancel the common factor of .
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Step 1.4.1.2.1.1.3.1
Cancel the common factor.
Step 1.4.1.2.1.1.3.2
Rewrite the expression.
Step 1.4.1.2.1.1.4
Raising to any positive power yields .
Step 1.4.1.2.1.2
Multiply by .
Step 1.4.1.2.1.3
Divide by .
Step 1.4.1.2.1.4
Simplify the numerator.
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Step 1.4.1.2.1.4.1
Rewrite as .
Step 1.4.1.2.1.4.2
Apply the power rule and multiply exponents, .
Step 1.4.1.2.1.4.3
Cancel the common factor of .
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Step 1.4.1.2.1.4.3.1
Cancel the common factor.
Step 1.4.1.2.1.4.3.2
Rewrite the expression.
Step 1.4.1.2.1.4.4
Raising to any positive power yields .
Step 1.4.1.2.1.5
Multiply by .
Step 1.4.1.2.1.6
Divide by .
Step 1.4.1.2.1.7
Multiply by .
Step 1.4.1.2.1.8
Raising to any positive power yields .
Step 1.4.1.2.1.9
Divide by .
Step 1.4.1.2.2
Simplify by adding and subtracting.
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Step 1.4.1.2.2.1
Add and .
Step 1.4.1.2.2.2
Add and .
Step 1.4.1.2.2.3
Subtract from .
Step 1.4.2
List all of the points.
Step 2
Evaluate at the included endpoints.
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Step 2.1
Evaluate at .
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Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
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Step 2.1.2.1
Simplify each term.
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Step 2.1.2.1.1
Simplify the numerator.
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Step 2.1.2.1.1.1
Rewrite as .
Step 2.1.2.1.1.2
Apply the power rule and multiply exponents, .
Step 2.1.2.1.1.3
Cancel the common factor of .
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Step 2.1.2.1.1.3.1
Cancel the common factor.
Step 2.1.2.1.1.3.2
Rewrite the expression.
Step 2.1.2.1.1.4
Raising to any positive power yields .
Step 2.1.2.1.2
Multiply by .
Step 2.1.2.1.3
Divide by .
Step 2.1.2.1.4
Simplify the numerator.
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Step 2.1.2.1.4.1
Rewrite as .
Step 2.1.2.1.4.2
Apply the power rule and multiply exponents, .
Step 2.1.2.1.4.3
Cancel the common factor of .
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Step 2.1.2.1.4.3.1
Cancel the common factor.
Step 2.1.2.1.4.3.2
Rewrite the expression.
Step 2.1.2.1.4.4
Raising to any positive power yields .
Step 2.1.2.1.5
Multiply by .
Step 2.1.2.1.6
Divide by .
Step 2.1.2.1.7
Multiply by .
Step 2.1.2.1.8
Raising to any positive power yields .
Step 2.1.2.1.9
Divide by .
Step 2.1.2.2
Simplify by adding and subtracting.
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Step 2.1.2.2.1
Add and .
Step 2.1.2.2.2
Add and .
Step 2.1.2.2.3
Subtract from .
Step 2.2
Evaluate at .
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Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
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Step 2.2.2.1
Simplify each term.
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Step 2.2.2.1.1
Simplify the numerator.
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Step 2.2.2.1.1.1
Rewrite as .
Step 2.2.2.1.1.2
Multiply the exponents in .
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Step 2.2.2.1.1.2.1
Apply the power rule and multiply exponents, .
Step 2.2.2.1.1.2.2
Cancel the common factor of .
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Step 2.2.2.1.1.2.2.1
Cancel the common factor.
Step 2.2.2.1.1.2.2.2
Rewrite the expression.
Step 2.2.2.1.1.3
Use the power rule to combine exponents.
Step 2.2.2.1.1.4
Add and .
Step 2.2.2.1.2
Raise to the power of .
Step 2.2.2.1.3
Multiply by by adding the exponents.
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Step 2.2.2.1.3.1
Multiply by .
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Step 2.2.2.1.3.1.1
Raise to the power of .
Step 2.2.2.1.3.1.2
Use the power rule to combine exponents.
Step 2.2.2.1.3.2
Write as a fraction with a common denominator.
Step 2.2.2.1.3.3
Combine the numerators over the common denominator.
Step 2.2.2.1.3.4
Add and .
Step 2.2.2.1.4
Simplify the numerator.
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Step 2.2.2.1.4.1
Rewrite as .
Step 2.2.2.1.4.2
Apply the power rule and multiply exponents, .
Step 2.2.2.1.4.3
Cancel the common factor of .
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Step 2.2.2.1.4.3.1
Cancel the common factor.
Step 2.2.2.1.4.3.2
Rewrite the expression.
Step 2.2.2.1.4.4
Raise to the power of .
Step 2.2.2.1.5
Raise to the power of .
Step 2.2.2.1.6
Divide by .
Step 2.2.2.2
Find the common denominator.
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Step 2.2.2.2.1
Multiply by .
Step 2.2.2.2.2
Multiply by .
Step 2.2.2.2.3
Multiply by .
Step 2.2.2.2.4
Multiply by .
Step 2.2.2.2.5
Write as a fraction with denominator .
Step 2.2.2.2.6
Multiply by .
Step 2.2.2.2.7
Multiply by .
Step 2.2.2.2.8
Write as a fraction with denominator .
Step 2.2.2.2.9
Multiply by .
Step 2.2.2.2.10
Multiply by .
Step 2.2.2.2.11
Reorder the factors of .
Step 2.2.2.2.12
Multiply by .
Step 2.2.2.2.13
Multiply by .
Step 2.2.2.3
Combine the numerators over the common denominator.
Step 2.2.2.4
Simplify each term.
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Step 2.2.2.4.1
Multiply by .
Step 2.2.2.4.2
Multiply by .
Step 2.2.2.4.3
Multiply by .
Step 2.2.2.4.4
Multiply by .
Step 2.2.2.5
Simplify by adding and subtracting.
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Step 2.2.2.5.1
Subtract from .
Step 2.2.2.5.2
Add and .
Step 2.2.2.5.3
Subtract from .
Step 2.3
List all of the points.
Step 3
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
Absolute Minimum:
Step 4