# Find the Local Maxima and Minima y=1/3x^3-5/2x^2+6x-3

Write as a function.
Find the first derivative of the function.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Combine and .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Combine and .
Multiply by .
Combine and .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
Find the second derivative of the function.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Differentiate using the Constant Rule.
Since is constant with respect to , the derivative of with respect to is .
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Factor using the AC method.
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Write the factored form using these integers.
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 .
Add to both sides of the equation.
Set the next factor equal to and solve.
Set the next factor equal to .
Add to both sides of the equation.
The final solution is all the values that make true.
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
Multiply by .
Subtract from .
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Find the y-value when .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Raise to the power of .
Multiply .
Multiply by .
Combine and .
Multiply by .
Move the negative in front of the fraction.
Multiply by .
Find the common denominator.
Write as a fraction with denominator .
Multiply by .
Multiply and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Combine fractions.
Combine fractions with similar denominators.
Multiply.
Multiply by .
Multiply by .
Multiply by .
Simplify the numerator.
Subtract from .
Subtract from .
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Evaluate the second derivative.
Multiply by .
Subtract from .
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Find the y-value when .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Combine and .
Raise to the power of .
Cancel the common factor of .
Move the leading negative in into the numerator.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply by .
Find the common denominator.
Write as a fraction with denominator .
Multiply by .
Multiply and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Combine fractions.
Combine fractions with similar denominators.
Multiply.
Multiply by .
Multiply by .
Multiply by .
Simplify the numerator.
Subtract from .