Calculus Examples

Find the Critical Points x^2e^x
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Differentiate using the Product Rule which states that is where and .
Step 1.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Simplify.
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Step 1.1.4.1
Reorder terms.
Step 1.1.4.2
Reorder factors in .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Factor out of .
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Step 2.2.1
Factor out of .
Step 2.2.2
Factor out of .
Step 2.2.3
Factor out of .
Step 2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.4
Set equal to .
Step 2.5
Set equal to and solve for .
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Step 2.5.1
Set equal to .
Step 2.5.2
Solve for .
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Step 2.5.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 2.5.2.2
The equation cannot be solved because is undefined.
Undefined
Step 2.5.2.3
There is no solution for
No solution
No solution
No solution
Step 2.6
Set equal to and solve for .
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Step 2.6.1
Set equal to .
Step 2.6.2
Subtract from both sides of the equation.
Step 2.7
The final solution is all the values that make true.
Step 3
Find the values where the derivative is undefined.
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Step 3.1
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 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
Raising to any positive power yields .
Step 4.1.2.2
Anything raised to is .
Step 4.1.2.3
Multiply by .
Step 4.2
Evaluate at .
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Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
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Step 4.2.2.1
Raise to the power of .
Step 4.2.2.2
Rewrite the expression using the negative exponent rule .
Step 4.2.2.3
Combine and .
Step 4.3
List all of the points.
Step 5