Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^3-3/2x^2-6x , [-2,3]
,
Step 1
Find the critical points.
Tap for more steps...
Step 1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1
Find the first derivative.
Tap for more steps...
Step 1.1.1.1
Differentiate.
Tap for more steps...
Step 1.1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2
Evaluate .
Tap for more steps...
Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Multiply by .
Step 1.1.1.2.4
Combine and .
Step 1.1.1.2.5
Multiply by .
Step 1.1.1.2.6
Combine and .
Step 1.1.1.2.7
Cancel the common factor of and .
Tap for more steps...
Step 1.1.1.2.7.1
Factor out of .
Step 1.1.1.2.7.2
Cancel the common factors.
Tap for more steps...
Step 1.1.1.2.7.2.1
Factor out of .
Step 1.1.1.2.7.2.2
Cancel the common factor.
Step 1.1.1.2.7.2.3
Rewrite the expression.
Step 1.1.1.2.7.2.4
Divide by .
Step 1.1.1.3
Evaluate .
Tap for more steps...
Step 1.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.3
Multiply by .
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
Tap for more steps...
Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Factor the left side of the equation.
Tap for more steps...
Step 1.2.2.1
Factor out of .
Tap for more steps...
Step 1.2.2.1.1
Factor out of .
Step 1.2.2.1.2
Factor out of .
Step 1.2.2.1.3
Factor out of .
Step 1.2.2.1.4
Factor out of .
Step 1.2.2.1.5
Factor out of .
Step 1.2.2.2
Factor.
Tap for more steps...
Step 1.2.2.2.1
Factor using the AC method.
Tap for more steps...
Step 1.2.2.2.1.1
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 .
Step 1.2.2.2.1.2
Write the factored form using these integers.
Step 1.2.2.2.2
Remove unnecessary parentheses.
Step 1.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.2.4
Set equal to and solve for .
Tap for more steps...
Step 1.2.4.1
Set equal to .
Step 1.2.4.2
Add to both sides of the equation.
Step 1.2.5
Set equal to and solve for .
Tap for more steps...
Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Subtract from both sides of the equation.
Step 1.2.6
The final solution is all the values that make true.
Step 1.3
Find the values where the derivative is undefined.
Tap for more steps...
Step 1.3.1
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 1.4
Evaluate at each value where the derivative is or undefined.
Tap for more steps...
Step 1.4.1
Evaluate at .
Tap for more steps...
Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
Tap for more steps...
Step 1.4.1.2.1
Simplify each term.
Tap for more steps...
Step 1.4.1.2.1.1
Raise to the power of .
Step 1.4.1.2.1.2
Cancel the common factor of .
Tap for more steps...
Step 1.4.1.2.1.2.1
Move the leading negative in into the numerator.
Step 1.4.1.2.1.2.2
Factor out of .
Step 1.4.1.2.1.2.3
Cancel the common factor.
Step 1.4.1.2.1.2.4
Rewrite the expression.
Step 1.4.1.2.1.3
Multiply by .
Step 1.4.1.2.1.4
Multiply by .
Step 1.4.1.2.2
Simplify by subtracting numbers.
Tap for more steps...
Step 1.4.1.2.2.1
Subtract from .
Step 1.4.1.2.2.2
Subtract from .
Step 1.4.2
Evaluate at .
Tap for more steps...
Step 1.4.2.1
Substitute for .
Step 1.4.2.2
Simplify.
Tap for more steps...
Step 1.4.2.2.1
Simplify each term.
Tap for more steps...
Step 1.4.2.2.1.1
Raise to the power of .
Step 1.4.2.2.1.2
Multiply by by adding the exponents.
Tap for more steps...
Step 1.4.2.2.1.2.1
Move .
Step 1.4.2.2.1.2.2
Multiply by .
Tap for more steps...
Step 1.4.2.2.1.2.2.1
Raise to the power of .
Step 1.4.2.2.1.2.2.2
Use the power rule to combine exponents.
Step 1.4.2.2.1.2.3
Add and .
Step 1.4.2.2.1.3
Raise to the power of .
Step 1.4.2.2.1.4
Multiply by .
Step 1.4.2.2.2
Find the common denominator.
Tap for more steps...
Step 1.4.2.2.2.1
Write as a fraction with denominator .
Step 1.4.2.2.2.2
Multiply by .
Step 1.4.2.2.2.3
Multiply by .
Step 1.4.2.2.2.4
Write as a fraction with denominator .
Step 1.4.2.2.2.5
Multiply by .
Step 1.4.2.2.2.6
Multiply by .
Step 1.4.2.2.3
Combine the numerators over the common denominator.
Step 1.4.2.2.4
Simplify each term.
Tap for more steps...
Step 1.4.2.2.4.1
Multiply by .
Step 1.4.2.2.4.2
Multiply by .
Step 1.4.2.2.5
Simplify by adding and subtracting.
Tap for more steps...
Step 1.4.2.2.5.1
Subtract from .
Step 1.4.2.2.5.2
Add and .
Step 1.4.3
List all of the points.
Step 2
Evaluate at the included endpoints.
Tap for more steps...
Step 2.1
Evaluate at .
Tap for more steps...
Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
Tap for more steps...
Step 2.1.2.1
Simplify each term.
Tap for more steps...
Step 2.1.2.1.1
Raise to the power of .
Step 2.1.2.1.2
Raise to the power of .
Step 2.1.2.1.3
Cancel the common factor of .
Tap for more steps...
Step 2.1.2.1.3.1
Move the leading negative in into the numerator.
Step 2.1.2.1.3.2
Factor out of .
Step 2.1.2.1.3.3
Cancel the common factor.
Step 2.1.2.1.3.4
Rewrite the expression.
Step 2.1.2.1.4
Multiply by .
Step 2.1.2.1.5
Multiply by .
Step 2.1.2.2
Simplify by adding and subtracting.
Tap for more steps...
Step 2.1.2.2.1
Subtract from .
Step 2.1.2.2.2
Add and .
Step 2.2
Evaluate at .
Tap for more steps...
Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
Tap for more steps...
Step 2.2.2.1
Simplify each term.
Tap for more steps...
Step 2.2.2.1.1
Raise to the power of .
Step 2.2.2.1.2
Raise to the power of .
Step 2.2.2.1.3
Multiply .
Tap for more steps...
Step 2.2.2.1.3.1
Multiply by .
Step 2.2.2.1.3.2
Combine and .
Step 2.2.2.1.3.3
Multiply by .
Step 2.2.2.1.4
Move the negative in front of the fraction.
Step 2.2.2.1.5
Multiply by .
Step 2.2.2.2
Find the common denominator.
Tap for more steps...
Step 2.2.2.2.1
Write as a fraction with denominator .
Step 2.2.2.2.2
Multiply by .
Step 2.2.2.2.3
Multiply by .
Step 2.2.2.2.4
Write as a fraction with denominator .
Step 2.2.2.2.5
Multiply by .
Step 2.2.2.2.6
Multiply by .
Step 2.2.2.3
Combine the numerators over the common denominator.
Step 2.2.2.4
Simplify each term.
Tap for more steps...
Step 2.2.2.4.1
Multiply by .
Step 2.2.2.4.2
Multiply by .
Step 2.2.2.5
Simplify the expression.
Tap for more steps...
Step 2.2.2.5.1
Subtract from .
Step 2.2.2.5.2
Subtract from .
Step 2.2.2.5.3
Move the negative in front of the fraction.
Step 2.3
List all of the points.
Step 3
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
Absolute Minimum:
Step 4