Calculus Examples

Find the Critical Points f(x)=3x^5-20x^3
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 1.2
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
Rewrite as .
Step 2.2.2
Let . Substitute for all occurrences of .
Step 2.2.3
Factor out of .
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Step 2.2.3.1
Factor out of .
Step 2.2.3.2
Factor out of .
Step 2.2.3.3
Factor out of .
Step 2.2.4
Replace all occurrences of with .
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
Add to both sides of the equation.
Step 2.5.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.5.2.3
Simplify .
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Step 2.5.2.3.1
Rewrite as .
Step 2.5.2.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.5.2.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 2.5.2.4.1
First, use the positive value of the to find the first solution.
Step 2.5.2.4.2
Next, use the negative value of the to find the second solution.
Step 2.5.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Add and .
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
Raise to the power of .
Step 4.2.2.1.2
Multiply by .
Step 4.2.2.1.3
Raise to the power of .
Step 4.2.2.1.4
Multiply by .
Step 4.2.2.2
Subtract from .
Step 4.3
Evaluate at .
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Step 4.3.1
Substitute for .
Step 4.3.2
Simplify.
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Step 4.3.2.1
Simplify each term.
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Step 4.3.2.1.1
Raise to the power of .
Step 4.3.2.1.2
Multiply by .
Step 4.3.2.1.3
Raise to the power of .
Step 4.3.2.1.4
Multiply by .
Step 4.3.2.2
Add and .
Step 4.4
List all of the points.
Step 5