Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=|x| , given -8<=x<=7
, given
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
The derivative of with respect to is .
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
Set the numerator equal to zero.
Step 1.2.3
Exclude the solutions that do not make true.
Step 1.3
Find the values where the derivative is undefined.
Tap for more steps...
Step 1.3.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.3.2
Solve for .
Tap for more steps...
Step 1.3.2.1
Remove the absolute value term. This creates a on the right side of the equation because .
Step 1.3.2.2
Plus or minus is .
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
The absolute value is the distance between a number and zero. The distance between and is .
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.2
Evaluate at .
Tap for more steps...
Step 2.2.1
Substitute for .
Step 2.2.2
The absolute value is the distance between a number and zero. The distance between and is .
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