Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=xe^(-2x)
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 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.3.5
Multiply by .
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
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Differentiate using the chain rule, which states that is where and .
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Step 2.3.1.1
To apply the Chain Rule, set as .
Step 2.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.1.3
Replace all occurrences of with .
Step 2.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.3
Differentiate using the Power Rule which states that is where .
Step 2.3.4
Multiply by .
Step 2.3.5
Move to the left of .
Step 2.4
Simplify.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Combine terms.
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Step 2.4.2.1
Multiply by .
Step 2.4.2.2
Subtract from .
Step 2.4.3
Reorder terms.
Step 2.4.4
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 4.1.3.4
Differentiate using the Power Rule which states that is where .
Step 4.1.3.5
Multiply by .
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
Multiply by .
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 natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 5.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 5.4.2.3
There is no solution for
No solution
No solution
No solution
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
Subtract from both sides of the equation.
Step 5.5.2.2
Divide each term in by and simplify.
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Step 5.5.2.2.1
Divide each term in by .
Step 5.5.2.2.2
Simplify the left side.
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Step 5.5.2.2.2.1
Cancel the common factor of .
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Step 5.5.2.2.2.1.1
Cancel the common factor.
Step 5.5.2.2.2.1.2
Divide by .
Step 5.5.2.2.3
Simplify the right side.
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Step 5.5.2.2.3.1
Dividing two negative values results in a positive value.
Step 5.6
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
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
Cancel the common factor of .
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Step 9.1.2.1
Factor out of .
Step 9.1.2.2
Cancel the common factor.
Step 9.1.2.3
Rewrite the expression.
Step 9.1.3
Rewrite the expression using the negative exponent rule .
Step 9.1.4
Combine and .
Step 9.1.5
Cancel the common factor of .
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Step 9.1.5.1
Factor out of .
Step 9.1.5.2
Cancel the common factor.
Step 9.1.5.3
Rewrite the expression.
Step 9.1.6
Rewrite the expression using the negative exponent rule .
Step 9.1.7
Combine and .
Step 9.1.8
Move the negative in front of the fraction.
Step 9.2
Combine fractions.
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Step 9.2.1
Combine the numerators over the common denominator.
Step 9.2.2
Simplify the expression.
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Step 9.2.2.1
Subtract from .
Step 9.2.2.2
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
Cancel the common factor of .
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Step 11.2.1.1
Factor out of .
Step 11.2.1.2
Cancel the common factor.
Step 11.2.1.3
Rewrite the expression.
Step 11.2.2
Rewrite the expression using the negative exponent rule .
Step 11.2.3
Multiply by .
Step 11.2.4
The final answer is .
Step 12
These are the local extrema for .
is a local maxima
Step 13