# Find the Inflection Points 1+1/x-1/(x^2)

Write as a function.
Find the second derivative.
Find the first derivative.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Evaluate .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Evaluate .
Differentiate using the Product Rule which states that is where and .
Rewrite as .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Subtract from .
Multiply by .
Multiply by .
Rewrite the expression using the negative exponent rule .
Rewrite the expression using the negative exponent rule .
Combine terms.
Subtract from .
Combine and .
Find the second derivative.
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Differentiate using the Product Rule which states that is where and .
Rewrite as .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Subtract from .
Multiply by .
Multiply by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Rewrite as .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Differentiate using the Power Rule which states that is where .
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Multiply by .
Multiply by .
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Subtract from .
Multiply by .
Rewrite the expression using the negative exponent rule .
Rewrite the expression using the negative exponent rule .
Combine terms.
Combine and .
Combine and .
Move the negative in front of the fraction.
The second derivative of with respect to is .
Set the second derivative equal to then solve the equation .
Set the second derivative equal to .
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Since contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.
The factors for are , which is multiplied by each other times.
occurs times.
The factors for are , which is multiplied by each other times.
occurs times.
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Simplify .
Multiply by .
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by by adding the exponents.
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply each term by and simplify.
Multiply each term in by in order to remove all the denominators from the equation.
Simplify each term.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Move the leading negative in into the numerator.
Cancel the common factor.
Rewrite the expression.
Multiply by .
Solve the equation.
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
Find the points where the second derivative is .
Substitute in to find the value of .
Replace the variable with in the expression.
Simplify the result.
Raise to the power of .
Find the common denominator.
Write as a fraction with denominator .
Multiply by .
Multiply and .
Multiply by .
Combine.
Multiply by .
Combine fractions.
Combine fractions with similar denominators.
Multiply by .
Simplify the numerator.
Subtract from .
The point found by substituting in is . This point can be an inflection point.
Split into intervals around the points that could potentially be inflection points.
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Divide by .
Raise to the power of .
Divide by .
Multiply by .
Subtract from .
At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval
Decreasing on since
Decreasing on since
Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raise to the power of .
Divide by .
Raise to the power of .
Divide by .
Multiply by .
Subtract from .