Calculus Examples

Find the Inflection Points f(x)=2x^3+3x^2-180x
Step 1
Find the second derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2
Evaluate .
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Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Multiply by .
Step 1.1.3
Evaluate .
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Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Multiply by .
Step 1.1.4
Evaluate .
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Step 1.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.2
Differentiate using the Power Rule which states that is where .
Step 1.1.4.3
Multiply by .
Step 1.2
Find the second derivative.
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Step 1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2.2
Evaluate .
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Step 1.2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.2.3
Multiply by .
Step 1.2.3
Evaluate .
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Step 1.2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.3.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3.3
Multiply by .
Step 1.2.4
Differentiate using the Constant Rule.
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Step 1.2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.4.2
Add and .
Step 1.3
The second derivative of with respect to is .
Step 2
Set the second derivative equal to then solve the equation .
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Step 2.1
Set the second derivative equal to .
Step 2.2
Subtract from both sides of the equation.
Step 2.3
Divide each term in by and simplify.
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Step 2.3.1
Divide each term in by .
Step 2.3.2
Simplify the left side.
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Step 2.3.2.1
Cancel the common factor of .
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Step 2.3.2.1.1
Cancel the common factor.
Step 2.3.2.1.2
Divide by .
Step 2.3.3
Simplify the right side.
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Step 2.3.3.1
Cancel the common factor of and .
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Step 2.3.3.1.1
Factor out of .
Step 2.3.3.1.2
Cancel the common factors.
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Step 2.3.3.1.2.1
Factor out of .
Step 2.3.3.1.2.2
Cancel the common factor.
Step 2.3.3.1.2.3
Rewrite the expression.
Step 2.3.3.2
Move the negative in front of the fraction.
Step 3
Find the points where the second derivative is .
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Step 3.1
Substitute in to find the value of .
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Step 3.1.1
Replace the variable with in the expression.
Step 3.1.2
Simplify the result.
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Step 3.1.2.1
Simplify each term.
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Step 3.1.2.1.1
Use the power rule to distribute the exponent.
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Step 3.1.2.1.1.1
Apply the product rule to .
Step 3.1.2.1.1.2
Apply the product rule to .
Step 3.1.2.1.2
Raise to the power of .
Step 3.1.2.1.3
One to any power is one.
Step 3.1.2.1.4
Raise to the power of .
Step 3.1.2.1.5
Cancel the common factor of .
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Step 3.1.2.1.5.1
Move the leading negative in into the numerator.
Step 3.1.2.1.5.2
Factor out of .
Step 3.1.2.1.5.3
Cancel the common factor.
Step 3.1.2.1.5.4
Rewrite the expression.
Step 3.1.2.1.6
Move the negative in front of the fraction.
Step 3.1.2.1.7
Use the power rule to distribute the exponent.
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Step 3.1.2.1.7.1
Apply the product rule to .
Step 3.1.2.1.7.2
Apply the product rule to .
Step 3.1.2.1.8
Raise to the power of .
Step 3.1.2.1.9
Multiply by .
Step 3.1.2.1.10
One to any power is one.
Step 3.1.2.1.11
Raise to the power of .
Step 3.1.2.1.12
Combine and .
Step 3.1.2.1.13
Cancel the common factor of .
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Step 3.1.2.1.13.1
Move the leading negative in into the numerator.
Step 3.1.2.1.13.2
Factor out of .
Step 3.1.2.1.13.3
Cancel the common factor.
Step 3.1.2.1.13.4
Rewrite the expression.
Step 3.1.2.1.14
Multiply by .
Step 3.1.2.2
Simplify terms.
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Step 3.1.2.2.1
Combine the numerators over the common denominator.
Step 3.1.2.2.2
Add and .
Step 3.1.2.2.3
Cancel the common factor of and .
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Step 3.1.2.2.3.1
Factor out of .
Step 3.1.2.2.3.2
Cancel the common factors.
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Step 3.1.2.2.3.2.1
Factor out of .
Step 3.1.2.2.3.2.2
Cancel the common factor.
Step 3.1.2.2.3.2.3
Rewrite the expression.
Step 3.1.2.3
To write as a fraction with a common denominator, multiply by .
Step 3.1.2.4
Combine and .
Step 3.1.2.5
Combine the numerators over the common denominator.
Step 3.1.2.6
Simplify the numerator.
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Step 3.1.2.6.1
Multiply by .
Step 3.1.2.6.2
Add and .
Step 3.1.2.7
The final answer is .
Step 3.2
The point found by substituting in is . This point can be an inflection point.
Step 4
Split into intervals around the points that could potentially be inflection points.
Step 5
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
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Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
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Step 5.2.1
Multiply by .
Step 5.2.2
Add and .
Step 5.2.3
The final answer is .
Step 5.3
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Step 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
Multiply by .
Step 6.2.2
Add and .
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
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 point in this case is .
Step 8