Calculus Examples

Find the Absolute Max and Min over the Interval -3x^4-8x^3-6x^2+1
Step 1
Find the first derivative of the function.
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Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
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Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
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Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Multiply by .
Step 1.4
Evaluate .
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Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.4.3
Multiply by .
Step 1.5
Differentiate using the Constant Rule.
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Step 1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.5.2
Add and .
Step 2
Find the second derivative of the function.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
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Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Evaluate .
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Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Multiply by .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Find the first derivative.
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Step 4.1
Find the first derivative.
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Step 4.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2
Evaluate .
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Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.1.3
Evaluate .
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Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Multiply by .
Step 4.1.4
Evaluate .
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Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Differentiate using the Power Rule which states that is where .
Step 4.1.4.3
Multiply by .
Step 4.1.5
Differentiate using the Constant Rule.
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Step 4.1.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.5.2
Add and .
Step 4.2
The first derivative of with respect to is .
Step 5
Set the first derivative equal to then solve the equation .
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Step 5.1
Set the first derivative equal to .
Step 5.2
Factor the left side of the equation.
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Step 5.2.1
Factor out of .
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Step 5.2.1.1
Factor out of .
Step 5.2.1.2
Factor out of .
Step 5.2.1.3
Factor out of .
Step 5.2.1.4
Factor out of .
Step 5.2.1.5
Factor out of .
Step 5.2.2
Factor using the perfect square rule.
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Step 5.2.2.1
Rewrite as .
Step 5.2.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 5.2.2.3
Rewrite the polynomial.
Step 5.2.2.4
Factor using the perfect square trinomial rule , where and .
Step 5.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4
Set equal to .
Step 5.5
Set equal to and solve for .
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Step 5.5.1
Set equal to .
Step 5.5.2
Solve for .
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Step 5.5.2.1
Set the equal to .
Step 5.5.2.2
Subtract from both sides of the equation.
Step 5.6
The final solution is all the values that make true.
Step 6
Find the values where the derivative is undefined.
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Step 6.1
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 7
Critical points to evaluate.
Step 8
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 9
Evaluate the second derivative.
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Step 9.1
Simplify each term.
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Step 9.1.1
Raising to any positive power yields .
Step 9.1.2
Multiply by .
Step 9.1.3
Multiply by .
Step 9.2
Simplify by adding and subtracting.
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Step 9.2.1
Add and .
Step 9.2.2
Subtract from .
Step 10
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 11
Find the y-value when .
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Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
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Step 11.2.1
Simplify each term.
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Step 11.2.1.1
Raising to any positive power yields .
Step 11.2.1.2
Multiply by .
Step 11.2.1.3
Raising to any positive power yields .
Step 11.2.1.4
Multiply by .
Step 11.2.1.5
Raising to any positive power yields .
Step 11.2.1.6
Multiply by .
Step 11.2.2
Simplify by adding numbers.
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Step 11.2.2.1
Add and .
Step 11.2.2.2
Add and .
Step 11.2.2.3
Add and .
Step 11.2.3
The final answer is .
Step 12
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 13
Evaluate the second derivative.
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Step 13.1
Simplify each term.
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Step 13.1.1
Raise to the power of .
Step 13.1.2
Multiply by .
Step 13.1.3
Multiply by .
Step 13.2
Simplify by adding and subtracting.
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Step 13.2.1
Add and .
Step 13.2.2
Subtract from .
Step 14
Since there is at least one point with or undefined second derivative, apply the first derivative test.
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Step 14.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 14.2
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 14.2.1
Replace the variable with in the expression.
Step 14.2.2
Simplify the result.
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Step 14.2.2.1
Simplify each term.
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Step 14.2.2.1.1
Raise to the power of .
Step 14.2.2.1.2
Multiply by .
Step 14.2.2.1.3
Raise to the power of .
Step 14.2.2.1.4
Multiply by .
Step 14.2.2.1.5
Multiply by .
Step 14.2.2.2
Simplify by adding and subtracting.
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Step 14.2.2.2.1
Subtract from .
Step 14.2.2.2.2
Add and .
Step 14.2.2.3
The final answer is .
Step 14.3
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 14.3.1
Replace the variable with in the expression.
Step 14.3.2
Simplify the result.
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Step 14.3.2.1
Simplify each term.
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Step 14.3.2.1.1
Raise to the power of .
Step 14.3.2.1.2
Multiply by .
Step 14.3.2.1.3
Raise to the power of .
Step 14.3.2.1.4
Multiply by .
Step 14.3.2.1.5
Multiply by .
Step 14.3.2.2
Simplify by adding and subtracting.
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Step 14.3.2.2.1
Subtract from .
Step 14.3.2.2.2
Add and .
Step 14.3.2.3
The final answer is .
Step 14.4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
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Step 14.4.1
Replace the variable with in the expression.
Step 14.4.2
Simplify the result.
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Step 14.4.2.1
Simplify each term.
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Step 14.4.2.1.1
Raise to the power of .
Step 14.4.2.1.2
Multiply by .
Step 14.4.2.1.3
Raise to the power of .
Step 14.4.2.1.4
Multiply by .
Step 14.4.2.1.5
Multiply by .
Step 14.4.2.2
Simplify by subtracting numbers.
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Step 14.4.2.2.1
Subtract from .
Step 14.4.2.2.2
Subtract from .
Step 14.4.2.3
The final answer is .
Step 14.5
Since the first derivative did not change signs around , this is not a local maximum or minimum.
Not a local maximum or minimum
Step 14.6
Since the first derivative changed signs from positive to negative around , then is a local maximum.
is a local maximum
is a local maximum
Step 15