Calculus Examples

Find the Inflection Points x^4-6x^2+5
Step 1
Write as a function.
Step 2
Find the second derivative.
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Step 2.1
Find the first derivative.
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Step 2.1.1
Differentiate.
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Step 2.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.1.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2
Evaluate .
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Step 2.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2.3
Multiply by .
Step 2.1.3
Differentiate using the Constant Rule.
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Step 2.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.3.2
Add and .
Step 2.2
Find the second derivative.
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Step 2.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.2
Evaluate .
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Step 2.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.2.3
Multiply by .
Step 2.2.3
Evaluate .
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Step 2.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3.3
Multiply by .
Step 2.3
The second derivative of with respect to is .
Step 3
Set the second derivative equal to then solve the equation .
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Step 3.1
Set the second derivative equal to .
Step 3.2
Add to both sides of the equation.
Step 3.3
Divide each term in by and simplify.
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Step 3.3.1
Divide each term in by .
Step 3.3.2
Simplify the left side.
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Step 3.3.2.1
Cancel the common factor of .
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Step 3.3.2.1.1
Cancel the common factor.
Step 3.3.2.1.2
Divide by .
Step 3.3.3
Simplify the right side.
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Step 3.3.3.1
Divide by .
Step 3.4
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.5
Any root of is .
Step 3.6
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.6.1
First, use the positive value of the to find the first solution.
Step 3.6.2
Next, use the negative value of the to find the second solution.
Step 3.6.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Find the points where the second derivative is .
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Step 4.1
Substitute in to find the value of .
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Step 4.1.1
Replace the variable with in the expression.
Step 4.1.2
Simplify the result.
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Step 4.1.2.1
Simplify each term.
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Step 4.1.2.1.1
One to any power is one.
Step 4.1.2.1.2
One to any power is one.
Step 4.1.2.1.3
Multiply by .
Step 4.1.2.2
Simplify by adding and subtracting.
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Step 4.1.2.2.1
Subtract from .
Step 4.1.2.2.2
Add and .
Step 4.1.2.3
The final answer is .
Step 4.2
The point found by substituting in is . This point can be an inflection point.
Step 4.3
Substitute in to find the value of .
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Step 4.3.1
Replace the variable with in the expression.
Step 4.3.2
Simplify the result.
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Step 4.3.2.1
Simplify each term.
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Step 4.3.2.1.1
Raise to the power of .
Step 4.3.2.1.2
Raise to the power of .
Step 4.3.2.1.3
Multiply by .
Step 4.3.2.2
Simplify by adding and subtracting.
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Step 4.3.2.2.1
Subtract from .
Step 4.3.2.2.2
Add and .
Step 4.3.2.3
The final answer is .
Step 4.4
The point found by substituting in is . This point can be an inflection point.
Step 4.5
Determine the points that could be inflection points.
Step 5
Split into intervals around the points that could potentially be inflection points.
Step 6
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
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Step 6.2.1
Simplify each term.
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Step 6.2.1.1
Raise to the power of .
Step 6.2.1.2
Multiply by .
Step 6.2.2
Subtract from .
Step 6.2.3
The final answer is .
Step 6.3
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Step 7
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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Step 7.1
Replace the variable with in the expression.
Step 7.2
Simplify the result.
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Step 7.2.1
Simplify each term.
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Step 7.2.1.1
Raising to any positive power yields .
Step 7.2.1.2
Multiply by .
Step 7.2.2
Subtract from .
Step 7.2.3
The final answer is .
Step 7.3
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Step 8
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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Step 8.1
Replace the variable with in the expression.
Step 8.2
Simplify the result.
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Step 8.2.1
Simplify each term.
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Step 8.2.1.1
Raise to the power of .
Step 8.2.1.2
Multiply by .
Step 8.2.2
Subtract from .
Step 8.2.3
The final answer is .
Step 8.3
At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .
Increasing on since
Increasing on since
Step 9
An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are .
Step 10