Calculus Examples

Find the Absolute Max and Min over the Interval g(x)=4x^3e^(-x) , -1<=x<=6
,
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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the Product Rule which states that is where and .
Step 1.1.1.3
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
Step 1.1.1.3.1
To apply the Chain Rule, set as .
Step 1.1.1.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.1.3.3
Replace all occurrences of with .
Step 1.1.1.4
Differentiate.
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
Differentiate using the Power Rule which states that is where .
Step 1.1.1.4.3
Simplify the expression.
Tap for more steps...
Step 1.1.1.4.3.1
Multiply by .
Step 1.1.1.4.3.2
Move to the left of .
Step 1.1.1.4.3.3
Rewrite as .
Step 1.1.1.4.4
Differentiate using the Power Rule which states that is where .
Step 1.1.1.5
Simplify.
Tap for more steps...
Step 1.1.1.5.1
Apply the distributive property.
Step 1.1.1.5.2
Combine terms.
Tap for more steps...
Step 1.1.1.5.2.1
Multiply by .
Step 1.1.1.5.2.2
Multiply by .
Step 1.1.1.5.3
Reorder terms.
Step 1.1.1.5.4
Reorder factors in .
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 and solve for .
Tap for more steps...
Step 1.2.4.1
Set equal to .
Step 1.2.4.2
Solve for .
Tap for more steps...
Step 1.2.4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.4.2.2
Simplify .
Tap for more steps...
Step 1.2.4.2.2.1
Rewrite as .
Step 1.2.4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2.4.2.2.3
Plus or minus is .
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
Solve for .
Tap for more steps...
Step 1.2.5.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 1.2.5.2.2
The equation cannot be solved because is undefined.
Undefined
Step 1.2.5.2.3
There is no solution for
No solution
No solution
No solution
Step 1.2.6
Set equal to and solve for .
Tap for more steps...
Step 1.2.6.1
Set equal to .
Step 1.2.6.2
Solve for .
Tap for more steps...
Step 1.2.6.2.1
Subtract from both sides of the equation.
Step 1.2.6.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 1.2.6.2.2.1
Divide each term in by .
Step 1.2.6.2.2.2
Simplify the left side.
Tap for more steps...
Step 1.2.6.2.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.6.2.2.2.2
Divide by .
Step 1.2.6.2.2.3
Simplify the right side.
Tap for more steps...
Step 1.2.6.2.2.3.1
Divide by .
Step 1.2.7
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
Raising to any positive power yields .
Step 1.4.1.2.2
Multiply by .
Step 1.4.1.2.3
Multiply by .
Step 1.4.1.2.4
Anything raised to is .
Step 1.4.1.2.5
Multiply by .
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
Raise to the power of .
Step 1.4.2.2.2
Multiply by .
Step 1.4.2.2.3
Multiply by .
Step 1.4.2.2.4
Rewrite the expression using the negative exponent rule .
Step 1.4.2.2.5
Combine 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
Raise to the power of .
Step 2.1.2.2
Multiply by .
Step 2.1.2.3
Multiply by .
Step 2.1.2.4
Simplify.
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
Raise to the power of .
Step 2.2.2.2
Multiply by .
Step 2.2.2.3
Multiply by .
Step 2.2.2.4
Rewrite the expression using the negative exponent rule .
Step 2.2.2.5
Combine 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