# Find the Critical Points x^8(x-3)^7 Find the derivative.
Differentiate using the Product Rule which states that is where and .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate.
Move to the left of .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Simplify the expression.
Multiply by .
Differentiate using the Power Rule which states that is where .
Simplify with factoring out.
Move to the left of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Set the derivative equal to .
Solve for .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Take the 7th root of both sides of the equation to eliminate the exponent on the left side.
Simplify .
Rewrite as .
Pull terms out from under the radical, assuming real numbers.
Set the next factor equal to and solve.
Set the next factor equal to .
Set the equal to .
Add to both sides of the equation.
Set the next factor equal to and solve.
Set the next factor equal to .
Simplify .
Simplify each term.
Apply the distributive property.
Multiply by .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
The final solution is all the values that make true.
Substitute the values of which cause the derivative to be into the original function.
Evaluate.
Raising to any positive power yields .
Subtract from .
Raise to the power of .
Multiply by .
Raise to the power of .
Subtract from .
Raising to any positive power yields .
Multiply by .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Use the power rule to distribute the exponent.
Apply the product rule to .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Raise to the power of .
Multiply .
Multiply and .
Multiply by .
Multiply by .
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.
Interval Notation:
Set-Builder Notation:
Since there are no values of where the derivative is undefined, there are no additional critical points.
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