Calculus Examples

Find the Critical Points f(x)=3x^4+6x^3-5
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
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 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 and solve for .
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Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
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Step 2.4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.4.2.2
Simplify .
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Step 2.4.2.2.1
Rewrite as .
Step 2.4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.4.2.2.3
Plus or minus is .
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
Subtract from both sides of the equation.
Step 2.5.2.2
Divide each term in by and simplify.
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Step 2.5.2.2.1
Divide each term in by .
Step 2.5.2.2.2
Simplify the left side.
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Step 2.5.2.2.2.1
Cancel the common factor of .
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Step 2.5.2.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.2.1.2
Divide by .
Step 2.5.2.2.3
Simplify the right side.
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Step 2.5.2.2.3.1
Move the negative in front of the fraction.
Step 2.6
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
Simplify each term.
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Step 4.1.2.1.1
Raising to any positive power yields .
Step 4.1.2.1.2
Multiply by .
Step 4.1.2.1.3
Raising to any positive power yields .
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
Add and .
Step 4.1.2.2.2
Subtract from .
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
Simplify each term.
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Step 4.2.2.1.1
Use the power rule to distribute the exponent.
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Step 4.2.2.1.1.1
Apply the product rule to .
Step 4.2.2.1.1.2
Apply the product rule to .
Step 4.2.2.1.2
Raise to the power of .
Step 4.2.2.1.3
Multiply by .
Step 4.2.2.1.4
Raise to the power of .
Step 4.2.2.1.5
Raise to the power of .
Step 4.2.2.1.6
Multiply .
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Step 4.2.2.1.6.1
Combine and .
Step 4.2.2.1.6.2
Multiply by .
Step 4.2.2.1.7
Use the power rule to distribute the exponent.
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Step 4.2.2.1.7.1
Apply the product rule to .
Step 4.2.2.1.7.2
Apply the product rule to .
Step 4.2.2.1.8
Raise to the power of .
Step 4.2.2.1.9
Raise to the power of .
Step 4.2.2.1.10
Raise to the power of .
Step 4.2.2.1.11
Cancel the common factor of .
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Step 4.2.2.1.11.1
Move the leading negative in into the numerator.
Step 4.2.2.1.11.2
Factor out of .
Step 4.2.2.1.11.3
Factor out of .
Step 4.2.2.1.11.4
Cancel the common factor.
Step 4.2.2.1.11.5
Rewrite the expression.
Step 4.2.2.1.12
Combine and .
Step 4.2.2.1.13
Multiply by .
Step 4.2.2.1.14
Move the negative in front of the fraction.
Step 4.2.2.2
Find the common denominator.
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Step 4.2.2.2.1
Multiply by .
Step 4.2.2.2.2
Multiply by .
Step 4.2.2.2.3
Write as a fraction with denominator .
Step 4.2.2.2.4
Multiply by .
Step 4.2.2.2.5
Multiply by .
Step 4.2.2.2.6
Multiply by .
Step 4.2.2.3
Combine the numerators over the common denominator.
Step 4.2.2.4
Simplify each term.
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Step 4.2.2.4.1
Multiply by .
Step 4.2.2.4.2
Multiply by .
Step 4.2.2.5
Simplify the expression.
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Step 4.2.2.5.1
Subtract from .
Step 4.2.2.5.2
Subtract from .
Step 4.2.2.5.3
Move the negative in front of the fraction.
Step 4.3
List all of the points.
Step 5