Calculus Examples

Find the Critical Points y=x^3-6x^2+12x-6
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.
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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
Differentiate using the Power Rule which states that is where .
Step 1.1.2
Evaluate .
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Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Multiply by .
Step 1.1.3
Evaluate .
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Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Differentiate using the Constant Rule.
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Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Add and .
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 the left side of the equation.
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Step 2.2.1
Factor out of .
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Step 2.2.1.1
Factor out of .
Step 2.2.1.2
Factor out of .
Step 2.2.1.3
Factor out of .
Step 2.2.1.4
Factor out of .
Step 2.2.1.5
Factor out of .
Step 2.2.2
Factor using the perfect square rule.
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Step 2.2.2.1
Rewrite as .
Step 2.2.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.2.2.3
Rewrite the polynomial.
Step 2.2.2.4
Factor using the perfect square trinomial rule , where and .
Step 2.3
Divide each term in by and simplify.
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Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Cancel the common factor of .
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Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
Divide by .
Step 2.4
Set the equal to .
Step 2.5
Add to both sides of the equation.
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
Simplify each term.
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Step 4.1.2.1.1
Raise to the power of .
Step 4.1.2.1.2
Raise to the power of .
Step 4.1.2.1.3
Multiply by .
Step 4.1.2.1.4
Multiply by .
Step 4.1.2.2
Simplify by adding and subtracting.
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Step 4.1.2.2.1
Subtract from .
Step 4.1.2.2.2
Add and .
Step 4.1.2.2.3
Subtract from .
Step 4.2
List all of the points.
Step 5