Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=5x^2-3x-1 , [-1,3]
,
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
By the Sum Rule, the derivative of with respect to is .
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 1.1.1.3
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 1.1.1.4
Differentiate using the Constant Rule.
Tap for more steps...
Step 1.1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.4.2
Add and .
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
Add to both sides of the equation.
Step 1.2.3
Divide each term in by and simplify.
Tap for more steps...
Step 1.2.3.1
Divide each term in by .
Step 1.2.3.2
Simplify the left side.
Tap for more steps...
Step 1.2.3.2.1
Cancel the common factor of .
Tap for more steps...
Step 1.2.3.2.1.1
Cancel the common factor.
Step 1.2.3.2.1.2
Divide by .
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
Apply the product rule to .
Step 1.4.1.2.1.2
Raise to the power of .
Step 1.4.1.2.1.3
Raise to the power of .
Step 1.4.1.2.1.4
Cancel the common factor of .
Tap for more steps...
Step 1.4.1.2.1.4.1
Factor out of .
Step 1.4.1.2.1.4.2
Cancel the common factor.
Step 1.4.1.2.1.4.3
Rewrite the expression.
Step 1.4.1.2.1.5
Multiply .
Tap for more steps...
Step 1.4.1.2.1.5.1
Combine and .
Step 1.4.1.2.1.5.2
Multiply by .
Step 1.4.1.2.1.6
Move the negative in front of the fraction.
Step 1.4.1.2.2
Find the common denominator.
Tap for more steps...
Step 1.4.1.2.2.1
Multiply by .
Step 1.4.1.2.2.2
Multiply by .
Step 1.4.1.2.2.3
Write as a fraction with denominator .
Step 1.4.1.2.2.4
Multiply by .
Step 1.4.1.2.2.5
Multiply by .
Step 1.4.1.2.2.6
Reorder the factors of .
Step 1.4.1.2.2.7
Multiply by .
Step 1.4.1.2.3
Combine the numerators over the common denominator.
Step 1.4.1.2.4
Simplify each term.
Tap for more steps...
Step 1.4.1.2.4.1
Multiply by .
Step 1.4.1.2.4.2
Multiply by .
Step 1.4.1.2.5
Simplify the expression.
Tap for more steps...
Step 1.4.1.2.5.1
Subtract from .
Step 1.4.1.2.5.2
Subtract from .
Step 1.4.1.2.5.3
Move the negative in front of the fraction.
Step 1.4.2
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
Multiply by .
Step 2.1.2.1.3
Multiply by .
Step 2.1.2.2
Simplify by adding and subtracting.
Tap for more steps...
Step 2.1.2.2.1
Add and .
Step 2.1.2.2.2
Subtract from .
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
Multiply by .
Step 2.2.2.1.3
Multiply by .
Step 2.2.2.2
Simplify by subtracting numbers.
Tap for more steps...
Step 2.2.2.2.1
Subtract from .
Step 2.2.2.2.2
Subtract from .
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