Calculus Examples

Find the Absolute Max and Min over the Interval g(x)=5x^3e^(-x) , -1<=x<=4
,
Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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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 .
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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.
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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.
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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.
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Step 1.1.1.5.1
Apply the distributive property.
Step 1.1.1.5.2
Combine terms.
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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 .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Factor out of .
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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 .
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Step 1.2.4.1
Set equal to .
Step 1.2.4.2
Solve for .
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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 .
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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 .
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Step 1.2.5.1
Set equal to .
Step 1.2.5.2
Solve for .
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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 .
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Step 1.2.6.1
Set equal to .
Step 1.2.6.2
Solve for .
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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.
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Step 1.2.6.2.2.1
Divide each term in by .
Step 1.2.6.2.2.2
Simplify the left side.
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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.
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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.
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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.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
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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 .
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Step 1.4.2.1
Substitute for .
Step 1.4.2.2
Simplify.
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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.
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Step 2.1
Evaluate at .
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Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
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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 .
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Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
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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