Calculus Examples

Find the Critical Points f(x)=2x^3+x^2+2x
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
By the Sum Rule, the derivative of with respect to is .
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
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Evaluate .
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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
Differentiate using the Power Rule which states that is where .
Step 1.1.4.3
Multiply by .
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 2.2.4
Factor out of .
Step 2.2.5
Factor out of .
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
Use the quadratic formula to find the solutions.
Step 2.5
Substitute the values , , and into the quadratic formula and solve for .
Step 2.6
Simplify.
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Step 2.6.1
Simplify the numerator.
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Step 2.6.1.1
One to any power is one.
Step 2.6.1.2
Multiply .
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Step 2.6.1.2.1
Multiply by .
Step 2.6.1.2.2
Multiply by .
Step 2.6.1.3
Subtract from .
Step 2.6.1.4
Rewrite as .
Step 2.6.1.5
Rewrite as .
Step 2.6.1.6
Rewrite as .
Step 2.6.2
Multiply by .
Step 2.7
Simplify the expression to solve for the portion of the .
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Step 2.7.1
Simplify the numerator.
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Step 2.7.1.1
One to any power is one.
Step 2.7.1.2
Multiply .
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Step 2.7.1.2.1
Multiply by .
Step 2.7.1.2.2
Multiply by .
Step 2.7.1.3
Subtract from .
Step 2.7.1.4
Rewrite as .
Step 2.7.1.5
Rewrite as .
Step 2.7.1.6
Rewrite as .
Step 2.7.2
Multiply by .
Step 2.7.3
Change the to .
Step 2.7.4
Rewrite as .
Step 2.7.5
Factor out of .
Step 2.7.6
Factor out of .
Step 2.7.7
Move the negative in front of the fraction.
Step 2.8
Simplify the expression to solve for the portion of the .
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Step 2.8.1
Simplify the numerator.
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Step 2.8.1.1
One to any power is one.
Step 2.8.1.2
Multiply .
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Step 2.8.1.2.1
Multiply by .
Step 2.8.1.2.2
Multiply by .
Step 2.8.1.3
Subtract from .
Step 2.8.1.4
Rewrite as .
Step 2.8.1.5
Rewrite as .
Step 2.8.1.6
Rewrite as .
Step 2.8.2
Multiply by .
Step 2.8.3
Change the to .
Step 2.8.4
Rewrite as .
Step 2.8.5
Factor out of .
Step 2.8.6
Factor out of .
Step 2.8.7
Move the negative in front of the fraction.
Step 2.9
The final answer is the combination of both solutions.
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
There are no values of in the domain of the original problem where the derivative is or undefined.
No critical points found