Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x^2-6x , (3,7]
,
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 1.1.2
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 1.2.3.3
Simplify the right side.
Tap for more steps...
Step 1.2.3.3.1
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
Raise to the power of .
Step 1.4.1.2.1.2
Multiply by .
Step 1.4.1.2.2
Subtract from .
Step 1.4.2
List all of the points.
Step 2
Exclude the points that are not on the interval.
Step 3
Evaluate at the included endpoints.
Tap for more steps...
Step 3.1
Evaluate at .
Tap for more steps...
Step 3.1.1
Substitute for .
Step 3.1.2
Simplify.
Tap for more steps...
Step 3.1.2.1
Simplify each term.
Tap for more steps...
Step 3.1.2.1.1
Raise to the power of .
Step 3.1.2.1.2
Multiply by .
Step 3.1.2.2
Subtract from .
Step 3.2
List all of the points.
Step 4
Since there is no value of that makes the first derivative equal to , there are no local extrema.
No Local Extrema
Step 5
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:
No absolute minimum
Step 6