Calculus Examples

Find the Absolute Max and Min over the Interval h(x)=x^3e^(-x)
Step 1
Find the first derivative of the function.
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Step 1.1
Differentiate using the Product Rule which states that is where and .
Step 1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.1
To apply the Chain Rule, set as .
Step 1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.2.3
Replace all occurrences of with .
Step 1.3
Differentiate.
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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
Simplify the expression.
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Step 1.3.3.1
Multiply by .
Step 1.3.3.2
Move to the left of .
Step 1.3.3.3
Rewrite as .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.4
Simplify.
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Step 1.4.1
Reorder terms.
Step 1.4.2
Reorder factors in .
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 Product Rule which states that is where and .
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 Exponential Rule which states that is where =.
Step 2.2.3.3
Replace all occurrences of with .
Step 2.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.5
Differentiate using the Power Rule which states that is where .
Step 2.2.6
Differentiate using the Power Rule which states that is where .
Step 2.2.7
Multiply by .
Step 2.2.8
Move to the left of .
Step 2.2.9
Rewrite as .
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 Product Rule which states that is where and .
Step 2.3.3
Differentiate using the chain rule, which states that is where and .
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Step 2.3.3.1
To apply the Chain Rule, set as .
Step 2.3.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.3.3
Replace all occurrences of with .
Step 2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.5
Differentiate using the Power Rule which states that is where .
Step 2.3.6
Differentiate using the Power Rule which states that is where .
Step 2.3.7
Multiply by .
Step 2.3.8
Move to the left of .
Step 2.3.9
Rewrite as .
Step 2.4
Simplify.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Apply the distributive property.
Step 2.4.3
Combine terms.
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Step 2.4.3.1
Multiply by .
Step 2.4.3.2
Multiply by .
Step 2.4.3.3
Multiply by .
Step 2.4.3.4
Multiply by .
Step 2.4.3.5
Multiply by .
Step 2.4.3.6
Subtract from .
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Step 2.4.3.6.1
Move .
Step 2.4.3.6.2
Subtract from .
Step 2.4.4
Reorder terms.
Step 2.4.5
Reorder factors in .
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 Product Rule which states that is where and .
Step 4.1.2
Differentiate using the chain rule, which states that is where and .
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Step 4.1.2.1
To apply the Chain Rule, set as .
Step 4.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.1.2.3
Replace all occurrences of with .
Step 4.1.3
Differentiate.
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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
Simplify the expression.
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Step 4.1.3.3.1
Multiply by .
Step 4.1.3.3.2
Move to the left of .
Step 4.1.3.3.3
Rewrite as .
Step 4.1.3.4
Differentiate using the Power Rule which states that is where .
Step 4.1.4
Simplify.
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Step 4.1.4.1
Reorder terms.
Step 4.1.4.2
Reorder factors in .
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
Factor out of .
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Step 5.2.1
Factor out of .
Step 5.2.2
Factor out of .
Step 5.2.3
Factor out of .
Step 5.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4
Set equal to and solve for .
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Step 5.4.1
Set equal to .
Step 5.4.2
Solve for .
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Step 5.4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.4.2.2
Simplify .
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Step 5.4.2.2.1
Rewrite as .
Step 5.4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 5.4.2.2.3
Plus or minus is .
Step 5.5
Set equal to and solve for .
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Step 5.5.1
Set equal to .
Step 5.5.2
Solve for .
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Step 5.5.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 5.5.2.2
The equation cannot be solved because is undefined.
Undefined
Step 5.5.2.3
There is no solution for
No solution
No solution
No solution
Step 5.6
Set equal to and solve for .
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Step 5.6.1
Set equal to .
Step 5.6.2
Solve for .
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Step 5.6.2.1
Subtract from both sides of the equation.
Step 5.6.2.2
Divide each term in by and simplify.
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Step 5.6.2.2.1
Divide each term in by .
Step 5.6.2.2.2
Simplify the left side.
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Step 5.6.2.2.2.1
Dividing two negative values results in a positive value.
Step 5.6.2.2.2.2
Divide by .
Step 5.6.2.2.3
Simplify the right side.
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Step 5.6.2.2.3.1
Divide by .
Step 5.7
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
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
Raising to any positive power yields .
Step 9.1.2
Multiply by .
Step 9.1.3
Anything raised to is .
Step 9.1.4
Multiply by .
Step 9.1.5
Raising to any positive power yields .
Step 9.1.6
Multiply by .
Step 9.1.7
Multiply by .
Step 9.1.8
Anything raised to is .
Step 9.1.9
Multiply by .
Step 9.1.10
Multiply by .
Step 9.1.11
Multiply by .
Step 9.1.12
Anything raised to is .
Step 9.1.13
Multiply by .
Step 9.2
Simplify by adding numbers.
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Step 9.2.1
Add and .
Step 9.2.2
Add and .
Step 10
Since there is at least one point with or undefined second derivative, apply the first derivative test.
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Step 10.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 10.2
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 10.2.1
Replace the variable with in the expression.
Step 10.2.2
Simplify the result.
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Step 10.2.2.1
Simplify each term.
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Step 10.2.2.1.1
Raise to the power of .
Step 10.2.2.1.2
Multiply by .
Step 10.2.2.1.3
Multiply by .
Step 10.2.2.1.4
Raise to the power of .
Step 10.2.2.1.5
Multiply by .
Step 10.2.2.1.6
Multiply by .
Step 10.2.2.2
Add and .
Step 10.2.2.3
The final answer is .
Step 10.3
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 10.3.1
Replace the variable with in the expression.
Step 10.3.2
Simplify the result.
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Step 10.3.2.1
Simplify each term.
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Step 10.3.2.1.1
Raise to the power of .
Step 10.3.2.1.2
Multiply by .
Step 10.3.2.1.3
Multiply by .
Step 10.3.2.1.4
Rewrite the expression using the negative exponent rule .
Step 10.3.2.1.5
Combine and .
Step 10.3.2.1.6
Move the negative in front of the fraction.
Step 10.3.2.1.7
Raise to the power of .
Step 10.3.2.1.8
Multiply by .
Step 10.3.2.1.9
Multiply by .
Step 10.3.2.1.10
Rewrite the expression using the negative exponent rule .
Step 10.3.2.1.11
Combine and .
Step 10.3.2.2
Combine fractions.
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Step 10.3.2.2.1
Combine the numerators over the common denominator.
Step 10.3.2.2.2
Add and .
Step 10.3.2.3
The final answer is .
Step 10.4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 10.4.1
Replace the variable with in the expression.
Step 10.4.2
Simplify the result.
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Step 10.4.2.1
Simplify each term.
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Step 10.4.2.1.1
Raise to the power of .
Step 10.4.2.1.2
Multiply by .
Step 10.4.2.1.3
Multiply by .
Step 10.4.2.1.4
Rewrite the expression using the negative exponent rule .
Step 10.4.2.1.5
Combine and .
Step 10.4.2.1.6
Move the negative in front of the fraction.
Step 10.4.2.1.7
Raise to the power of .
Step 10.4.2.1.8
Multiply by .
Step 10.4.2.1.9
Multiply by .
Step 10.4.2.1.10
Rewrite the expression using the negative exponent rule .
Step 10.4.2.1.11
Combine and .
Step 10.4.2.2
Combine fractions.
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Step 10.4.2.2.1
Combine the numerators over the common denominator.
Step 10.4.2.2.2
Simplify the expression.
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Step 10.4.2.2.2.1
Add and .
Step 10.4.2.2.2.2
Move the negative in front of the fraction.
Step 10.4.2.3
The final answer is .
Step 10.5
Since the first derivative did not change signs around , this is not a local maximum or minimum.
Not a local maximum or minimum
Step 10.6
Since the first derivative changed signs from positive to negative around , then is a local maximum.
is a local maximum
is a local maximum
Step 11