Calculus Examples

Find the Critical Points f(x) = cube root of 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
Use to rewrite as .
Step 1.1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Replace all occurrences of with .
Step 1.1.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.4
Combine and .
Step 1.1.5
Combine the numerators over the common denominator.
Step 1.1.6
Simplify the numerator.
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Step 1.1.6.1
Multiply by .
Step 1.1.6.2
Subtract from .
Step 1.1.7
Combine fractions.
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Step 1.1.7.1
Move the negative in front of the fraction.
Step 1.1.7.2
Combine and .
Step 1.1.7.3
Move to the denominator using the negative exponent rule .
Step 1.1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.1.9
Differentiate using the Power Rule which states that is where .
Step 1.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.11
Differentiate using the Power Rule which states that is where .
Step 1.1.12
Multiply by .
Step 1.1.13
Simplify.
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Step 1.1.13.1
Reorder the factors of .
Step 1.1.13.2
Multiply by .
Step 1.1.13.3
Factor out of .
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Step 1.1.13.3.1
Factor out of .
Step 1.1.13.3.2
Factor out of .
Step 1.1.13.3.3
Factor out of .
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
Set the numerator equal to zero.
Step 2.3
Solve the equation for .
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Step 2.3.1
Divide each term in by and simplify.
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Step 2.3.1.1
Divide each term in by .
Step 2.3.1.2
Simplify the left side.
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Step 2.3.1.2.1
Cancel the common factor of .
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Step 2.3.1.2.1.1
Cancel the common factor.
Step 2.3.1.2.1.2
Divide by .
Step 2.3.1.3
Simplify the right side.
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Step 2.3.1.3.1
Divide by .
Step 2.3.2
Add to both sides of the equation.
Step 3
Find the values where the derivative is undefined.
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Step 3.1
Apply the rule to rewrite the exponentiation as a radical.
Step 3.2
Set the denominator in equal to to find where the expression is undefined.
Step 3.3
Solve for .
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Step 3.3.1
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 3.3.2
Simplify each side of the equation.
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Step 3.3.2.1
Use to rewrite as .
Step 3.3.2.2
Simplify the left side.
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Step 3.3.2.2.1
Simplify .
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Step 3.3.2.2.1.1
Apply the product rule to .
Step 3.3.2.2.1.2
Raise to the power of .
Step 3.3.2.2.1.3
Multiply the exponents in .
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Step 3.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 3.3.2.2.1.3.2
Cancel the common factor of .
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Step 3.3.2.2.1.3.2.1
Cancel the common factor.
Step 3.3.2.2.1.3.2.2
Rewrite the expression.
Step 3.3.2.3
Simplify the right side.
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Step 3.3.2.3.1
Raising to any positive power yields .
Step 3.3.3
Solve for .
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Step 3.3.3.1
Factor the left side of the equation.
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Step 3.3.3.1.1
Factor out of .
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Step 3.3.3.1.1.1
Factor out of .
Step 3.3.3.1.1.2
Factor out of .
Step 3.3.3.1.1.3
Factor out of .
Step 3.3.3.1.2
Factor.
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Step 3.3.3.1.2.1
Apply the product rule to .
Step 3.3.3.1.2.2
Remove unnecessary parentheses.
Step 3.3.3.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.3.3.3
Set equal to and solve for .
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Step 3.3.3.3.1
Set equal to .
Step 3.3.3.3.2
Solve for .
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Step 3.3.3.3.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.3.3.3.2.2
Simplify .
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Step 3.3.3.3.2.2.1
Rewrite as .
Step 3.3.3.3.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3.3.3.3.2.2.3
Plus or minus is .
Step 3.3.3.4
Set equal to and solve for .
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Step 3.3.3.4.1
Set equal to .
Step 3.3.3.4.2
Solve for .
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Step 3.3.3.4.2.1
Set the equal to .
Step 3.3.3.4.2.2
Add to both sides of the equation.
Step 3.3.3.5
The final solution is all the values that make true.
Step 3.4
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
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
One to any power is one.
Step 4.1.2.2
Multiply by .
Step 4.1.2.3
Subtract from .
Step 4.1.2.4
Rewrite as .
Step 4.1.2.5
Pull terms out from under the radical, assuming real numbers.
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
Raising to any positive power yields .
Step 4.2.2.2
Multiply by .
Step 4.2.2.3
Add and .
Step 4.2.2.4
Rewrite as .
Step 4.2.2.5
Pull terms out from under the radical, assuming real numbers.
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
Raise to the power of .
Step 4.3.2.2
Multiply by .
Step 4.3.2.3
Subtract from .
Step 4.3.2.4
Rewrite as .
Step 4.3.2.5
Pull terms out from under the radical, assuming real numbers.
Step 4.4
List all of the points.
Step 5