Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=-x^3+6x^2-10x+4
Step 1
Find the first derivative of the function.
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Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
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Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
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Step 1.3.1
Since is constant with respect to , 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
Multiply by .
Step 1.4
Evaluate .
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Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.4.3
Multiply by .
Step 1.5
Differentiate using the Constant Rule.
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Step 1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.5.2
Add and .
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
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Differentiate using the Constant Rule.
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Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
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
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2
Evaluate .
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Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.1.3
Evaluate .
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Step 4.1.3.1
Since is constant with respect to , 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
Multiply by .
Step 4.1.4
Evaluate .
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Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Differentiate using the Power Rule which states that is where .
Step 4.1.4.3
Multiply by .
Step 4.1.5
Differentiate using the Constant Rule.
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Step 4.1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5.2
Add 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
Use the quadratic formula to find the solutions.
Step 5.3
Substitute the values , , and into the quadratic formula and solve for .
Step 5.4
Simplify.
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Step 5.4.1
Simplify the numerator.
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Step 5.4.1.1
Raise to the power of .
Step 5.4.1.2
Multiply .
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Step 5.4.1.2.1
Multiply by .
Step 5.4.1.2.2
Multiply by .
Step 5.4.1.3
Subtract from .
Step 5.4.1.4
Rewrite as .
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Step 5.4.1.4.1
Factor out of .
Step 5.4.1.4.2
Rewrite as .
Step 5.4.1.5
Pull terms out from under the radical.
Step 5.4.2
Multiply by .
Step 5.4.3
Simplify .
Step 5.5
Simplify the expression to solve for the portion of the .
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Step 5.5.1
Simplify the numerator.
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Step 5.5.1.1
Raise to the power of .
Step 5.5.1.2
Multiply .
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Step 5.5.1.2.1
Multiply by .
Step 5.5.1.2.2
Multiply by .
Step 5.5.1.3
Subtract from .
Step 5.5.1.4
Rewrite as .
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Step 5.5.1.4.1
Factor out of .
Step 5.5.1.4.2
Rewrite as .
Step 5.5.1.5
Pull terms out from under the radical.
Step 5.5.2
Multiply by .
Step 5.5.3
Simplify .
Step 5.5.4
Change the to .
Step 5.6
Simplify the expression to solve for the portion of the .
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Step 5.6.1
Simplify the numerator.
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Step 5.6.1.1
Raise to the power of .
Step 5.6.1.2
Multiply .
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Step 5.6.1.2.1
Multiply by .
Step 5.6.1.2.2
Multiply by .
Step 5.6.1.3
Subtract from .
Step 5.6.1.4
Rewrite as .
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Step 5.6.1.4.1
Factor out of .
Step 5.6.1.4.2
Rewrite as .
Step 5.6.1.5
Pull terms out from under the radical.
Step 5.6.2
Multiply by .
Step 5.6.3
Simplify .
Step 5.6.4
Change the to .
Step 5.7
The final answer is the combination of both solutions.
Step 6
Find the values where the derivative is undefined.
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Step 6.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
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 each term.
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Step 9.1.1
Cancel the common factor of .
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Step 9.1.1.1
Factor out of .
Step 9.1.1.2
Cancel the common factor.
Step 9.1.1.3
Rewrite the expression.
Step 9.1.2
Apply the distributive property.
Step 9.1.3
Multiply by .
Step 9.2
Simplify by adding numbers.
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Step 9.2.1
Add and .
Step 9.2.2
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
Simplify each term.
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Step 11.2.1.1
Apply the product rule to .
Step 11.2.1.2
Raise to the power of .
Step 11.2.1.3
Use the Binomial Theorem.
Step 11.2.1.4
Simplify each term.
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Step 11.2.1.4.1
Raise to the power of .
Step 11.2.1.4.2
Raise to the power of .
Step 11.2.1.4.3
Multiply by .
Step 11.2.1.4.4
Multiply by .
Step 11.2.1.4.5
Rewrite as .
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Step 11.2.1.4.5.1
Use to rewrite as .
Step 11.2.1.4.5.2
Apply the power rule and multiply exponents, .
Step 11.2.1.4.5.3
Combine and .
Step 11.2.1.4.5.4
Cancel the common factor of .
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Step 11.2.1.4.5.4.1
Cancel the common factor.
Step 11.2.1.4.5.4.2
Rewrite the expression.
Step 11.2.1.4.5.5
Evaluate the exponent.
Step 11.2.1.4.6
Multiply by .
Step 11.2.1.4.7
Rewrite as .
Step 11.2.1.4.8
Raise to the power of .
Step 11.2.1.4.9
Rewrite as .
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Step 11.2.1.4.9.1
Factor out of .
Step 11.2.1.4.9.2
Rewrite as .
Step 11.2.1.4.10
Pull terms out from under the radical.
Step 11.2.1.5
Add and .
Step 11.2.1.6
Add and .
Step 11.2.1.7
Cancel the common factor of and .
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Step 11.2.1.7.1
Factor out of .
Step 11.2.1.7.2
Factor out of .
Step 11.2.1.7.3
Factor out of .
Step 11.2.1.7.4
Cancel the common factors.
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Step 11.2.1.7.4.1
Factor out of .
Step 11.2.1.7.4.2
Cancel the common factor.
Step 11.2.1.7.4.3
Rewrite the expression.
Step 11.2.1.8
Apply the product rule to .
Step 11.2.1.9
Raise to the power of .
Step 11.2.1.10
Cancel the common factor of .
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Step 11.2.1.10.1
Factor out of .
Step 11.2.1.10.2
Factor out of .
Step 11.2.1.10.3
Cancel the common factor.
Step 11.2.1.10.4
Rewrite the expression.
Step 11.2.1.11
Combine and .
Step 11.2.1.12
Rewrite as .
Step 11.2.1.13
Expand using the FOIL Method.
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Step 11.2.1.13.1
Apply the distributive property.
Step 11.2.1.13.2
Apply the distributive property.
Step 11.2.1.13.3
Apply the distributive property.
Step 11.2.1.14
Simplify and combine like terms.
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Step 11.2.1.14.1
Simplify each term.
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Step 11.2.1.14.1.1
Multiply by .
Step 11.2.1.14.1.2
Move to the left of .
Step 11.2.1.14.1.3
Combine using the product rule for radicals.
Step 11.2.1.14.1.4
Multiply by .
Step 11.2.1.14.1.5
Rewrite as .
Step 11.2.1.14.1.6
Pull terms out from under the radical, assuming positive real numbers.
Step 11.2.1.14.2
Add and .
Step 11.2.1.14.3
Add and .
Step 11.2.1.15
Cancel the common factor of and .
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Step 11.2.1.15.1
Factor out of .
Step 11.2.1.15.2
Cancel the common factors.
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Step 11.2.1.15.2.1
Factor out of .
Step 11.2.1.15.2.2
Cancel the common factor.
Step 11.2.1.15.2.3
Rewrite the expression.
Step 11.2.1.15.2.4
Divide by .
Step 11.2.1.16
Apply the distributive property.
Step 11.2.1.17
Multiply by .
Step 11.2.1.18
Multiply by .
Step 11.2.1.19
Combine and .
Step 11.2.1.20
Move the negative in front of the fraction.
Step 11.2.2
To write as a fraction with a common denominator, multiply by .
Step 11.2.3
Combine fractions.
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Step 11.2.3.1
Combine and .
Step 11.2.3.2
Simplify the expression.
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Step 11.2.3.2.1
Combine the numerators over the common denominator.
Step 11.2.3.2.2
Multiply by .
Step 11.2.4
Simplify the numerator.
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Step 11.2.4.1
Apply the distributive property.
Step 11.2.4.2
Multiply by .
Step 11.2.4.3
Multiply by .
Step 11.2.4.4
Add and .
Step 11.2.5
Find the common denominator.
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Step 11.2.5.1
Write as a fraction with denominator .
Step 11.2.5.2
Multiply by .
Step 11.2.5.3
Multiply by .
Step 11.2.5.4
Multiply by .
Step 11.2.5.5
Multiply by .
Step 11.2.5.6
Write as a fraction with denominator .
Step 11.2.5.7
Multiply by .
Step 11.2.5.8
Multiply by .
Step 11.2.5.9
Multiply by .
Step 11.2.6
Combine the numerators over the common denominator.
Step 11.2.7
Simplify each term.
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Step 11.2.7.1
Multiply by .
Step 11.2.7.2
Apply the distributive property.
Step 11.2.7.3
Multiply by .
Step 11.2.7.4
Apply the distributive property.
Step 11.2.7.5
Multiply by .
Step 11.2.7.6
Multiply by .
Step 11.2.7.7
Multiply by .
Step 11.2.8
Simplify by adding terms.
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Step 11.2.8.1
Subtract from .
Step 11.2.8.2
Simplify by adding numbers.
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Step 11.2.8.2.1
Add and .
Step 11.2.8.2.2
Subtract from .
Step 11.2.8.3
Add and .
Step 11.2.8.4
Subtract from .
Step 11.2.9
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 each term.
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Step 13.1.1
Cancel the common factor of .
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Step 13.1.1.1
Factor out of .
Step 13.1.1.2
Cancel the common factor.
Step 13.1.1.3
Rewrite the expression.
Step 13.1.2
Apply the distributive property.
Step 13.1.3
Multiply by .
Step 13.1.4
Multiply by .
Step 13.2
Simplify by adding numbers.
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Step 13.2.1
Add and .
Step 13.2.2
Add and .
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 each term.
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Step 15.2.1.1
Apply the product rule to .
Step 15.2.1.2
Raise to the power of .
Step 15.2.1.3
Use the Binomial Theorem.
Step 15.2.1.4
Simplify each term.
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Step 15.2.1.4.1
Raise to the power of .
Step 15.2.1.4.2
Raise to the power of .
Step 15.2.1.4.3
Multiply by .
Step 15.2.1.4.4
Multiply by .
Step 15.2.1.4.5
Multiply by .
Step 15.2.1.4.6
Apply the product rule to .
Step 15.2.1.4.7
Raise to the power of .
Step 15.2.1.4.8
Multiply by .
Step 15.2.1.4.9
Rewrite as .
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Step 15.2.1.4.9.1
Use to rewrite as .
Step 15.2.1.4.9.2
Apply the power rule and multiply exponents, .
Step 15.2.1.4.9.3
Combine and .
Step 15.2.1.4.9.4
Cancel the common factor of .
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Step 15.2.1.4.9.4.1
Cancel the common factor.
Step 15.2.1.4.9.4.2
Rewrite the expression.
Step 15.2.1.4.9.5
Evaluate the exponent.
Step 15.2.1.4.10
Multiply by .
Step 15.2.1.4.11
Apply the product rule to .
Step 15.2.1.4.12
Raise to the power of .
Step 15.2.1.4.13
Rewrite as .
Step 15.2.1.4.14
Raise to the power of .
Step 15.2.1.4.15
Rewrite as .
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Step 15.2.1.4.15.1
Factor out of .
Step 15.2.1.4.15.2
Rewrite as .
Step 15.2.1.4.16
Pull terms out from under the radical.
Step 15.2.1.4.17
Multiply by .
Step 15.2.1.5
Add and .
Step 15.2.1.6
Subtract from .
Step 15.2.1.7
Cancel the common factor of and .
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Step 15.2.1.7.1
Factor out of .
Step 15.2.1.7.2
Factor out of .
Step 15.2.1.7.3
Factor out of .
Step 15.2.1.7.4
Cancel the common factors.
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Step 15.2.1.7.4.1
Factor out of .
Step 15.2.1.7.4.2
Cancel the common factor.
Step 15.2.1.7.4.3
Rewrite the expression.
Step 15.2.1.8
Apply the product rule to .
Step 15.2.1.9
Raise to the power of .
Step 15.2.1.10
Cancel the common factor of .
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Step 15.2.1.10.1
Factor out of .
Step 15.2.1.10.2
Factor out of .
Step 15.2.1.10.3
Cancel the common factor.
Step 15.2.1.10.4
Rewrite the expression.
Step 15.2.1.11
Combine and .
Step 15.2.1.12
Rewrite as .
Step 15.2.1.13
Expand using the FOIL Method.
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Step 15.2.1.13.1
Apply the distributive property.
Step 15.2.1.13.2
Apply the distributive property.
Step 15.2.1.13.3
Apply the distributive property.
Step 15.2.1.14
Simplify and combine like terms.
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Step 15.2.1.14.1
Simplify each term.
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Step 15.2.1.14.1.1
Multiply by .
Step 15.2.1.14.1.2
Multiply by .
Step 15.2.1.14.1.3
Multiply by .
Step 15.2.1.14.1.4
Multiply .
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Step 15.2.1.14.1.4.1
Multiply by .
Step 15.2.1.14.1.4.2
Multiply by .
Step 15.2.1.14.1.4.3
Raise to the power of .
Step 15.2.1.14.1.4.4
Raise to the power of .
Step 15.2.1.14.1.4.5
Use the power rule to combine exponents.
Step 15.2.1.14.1.4.6
Add and .
Step 15.2.1.14.1.5
Rewrite as .
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Step 15.2.1.14.1.5.1
Use to rewrite as .
Step 15.2.1.14.1.5.2
Apply the power rule and multiply exponents, .
Step 15.2.1.14.1.5.3
Combine and .
Step 15.2.1.14.1.5.4
Cancel the common factor of .
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Step 15.2.1.14.1.5.4.1
Cancel the common factor.
Step 15.2.1.14.1.5.4.2
Rewrite the expression.
Step 15.2.1.14.1.5.5
Evaluate the exponent.
Step 15.2.1.14.2
Add and .
Step 15.2.1.14.3
Subtract from .
Step 15.2.1.15
Cancel the common factor of and .
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Step 15.2.1.15.1
Factor out of .
Step 15.2.1.15.2
Cancel the common factors.
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Step 15.2.1.15.2.1
Factor out of .
Step 15.2.1.15.2.2
Cancel the common factor.
Step 15.2.1.15.2.3
Rewrite the expression.
Step 15.2.1.15.2.4
Divide by .
Step 15.2.1.16
Apply the distributive property.
Step 15.2.1.17
Multiply by .
Step 15.2.1.18
Multiply by .
Step 15.2.1.19
Combine and .
Step 15.2.1.20
Move the negative in front of the fraction.
Step 15.2.2
To write as a fraction with a common denominator, multiply by .
Step 15.2.3
Combine fractions.
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Step 15.2.3.1
Combine and .
Step 15.2.3.2
Simplify the expression.
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Step 15.2.3.2.1
Combine the numerators over the common denominator.
Step 15.2.3.2.2
Multiply by .
Step 15.2.4
Simplify the numerator.
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Step 15.2.4.1
Apply the distributive property.
Step 15.2.4.2
Multiply by .
Step 15.2.4.3
Multiply by .
Step 15.2.4.4
Add and .
Step 15.2.5
Find the common denominator.
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Step 15.2.5.1
Write as a fraction with denominator .
Step 15.2.5.2
Multiply by .
Step 15.2.5.3
Multiply by .
Step 15.2.5.4
Multiply by .
Step 15.2.5.5
Multiply by .
Step 15.2.5.6
Write as a fraction with denominator .
Step 15.2.5.7
Multiply by .
Step 15.2.5.8
Multiply by .
Step 15.2.5.9
Multiply by .
Step 15.2.6
Combine the numerators over the common denominator.
Step 15.2.7
Simplify each term.
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Step 15.2.7.1
Multiply by .
Step 15.2.7.2
Apply the distributive property.
Step 15.2.7.3
Multiply by .
Step 15.2.7.4
Multiply by .
Step 15.2.7.5
Apply the distributive property.
Step 15.2.7.6
Multiply by .
Step 15.2.7.7
Multiply by .
Step 15.2.7.8
Multiply by .
Step 15.2.8
Simplify by adding terms.
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Step 15.2.8.1
Subtract from .
Step 15.2.8.2
Simplify by adding numbers.
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Step 15.2.8.2.1
Add and .
Step 15.2.8.2.2
Add and .
Step 15.2.8.3
Subtract from .
Step 15.2.8.4
Add and .
Step 15.2.8.5
Move the negative in front of the fraction.
Step 15.2.9
The final answer is .
Step 16
These are the local extrema for .
is a local maxima
is a local minima
Step 17