Calculus Examples

Find the Absolute Max and Min over the Interval h(x)=x^3+3x^2+5 on -3 , 2
on ,
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 1.1.1.3
Differentiate using the Constant Rule.
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
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
Factor out of .
Tap for more steps...
Step 1.2.2.1
Factor out of .
Step 1.2.2.2
Factor out of .
Step 1.2.2.3
Factor out of .
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 .
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
Raising to any positive power yields .
Step 1.4.1.2.1.2
Raising to any positive power yields .
Step 1.4.1.2.1.3
Multiply by .
Step 1.4.1.2.2
Simplify by adding numbers.
Tap for more steps...
Step 1.4.1.2.2.1
Add and .
Step 1.4.1.2.2.2
Add and .
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
Raise to the power of .
Step 1.4.2.2.1.3
Multiply by .
Step 1.4.2.2.2
Simplify by adding numbers.
Tap for more steps...
Step 1.4.2.2.2.1
Add and .
Step 1.4.2.2.2.2
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
Multiply by .
Step 2.1.2.2
Simplify by adding numbers.
Tap for more steps...
Step 2.1.2.2.1
Add and .
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 by .
Step 2.2.2.2
Simplify by adding numbers.
Tap for more steps...
Step 2.2.2.2.1
Add and .
Step 2.2.2.2.2
Add and .
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