Find The Inflection Points 5x^(2/3)-2x^(5/3) | Mathway

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Calculus Examples Popular Problems Calculus Find the Inflection Points 5x^(2/3)-2x^(5/3) Step 1Write as a function.Step 2Find the second derivative.Tap for more steps...Step 2.1Find the first derivative.Tap for more steps...Step 2.1.1By the Sum Rule, the derivative of with respect to is .Step 2.1.2Evaluate .Tap for more steps...Step 2.1.2.1Since is constant with respect to , the derivative of with respect to is .Step 2.1.2.2Differentiate using the Power Rule which states that is where .Step 2.1.2.3To write as a fraction with a common denominator, multiply by .Step 2.1.2.4Combine and .Step 2.1.2.5Combine the numerators over the common denominator.Step 2.1.2.6Simplify the numerator.Tap for more steps...Step 2.1.2.6.1Multiply by .Step 2.1.2.6.2Subtract from .Step 2.1.2.7Move the negative in front of the fraction.Step 2.1.2.8Combine and .Step 2.1.2.9Combine and .Step 2.1.2.10Multiply by .Step 2.1.2.11Move to the denominator using the negative exponent rule .Step 2.1.3Evaluate .Tap for more steps...Step 2.1.3.1Since is constant with respect to , the derivative of with respect to is .Step 2.1.3.2Differentiate using the Power Rule which states that is where .Step 2.1.3.3To write as a fraction with a common denominator, multiply by .Step 2.1.3.4Combine and .Step 2.1.3.5Combine the numerators over the common denominator.Step 2.1.3.6Simplify the numerator.Tap for more steps...Step 2.1.3.6.1Multiply by .Step 2.1.3.6.2Subtract from .Step 2.1.3.7Combine and .Step 2.1.3.8Combine and .Step 2.1.3.9Multiply by .Step 2.1.3.10Move the negative in front of the fraction.Step 2.2Find the second derivative.Tap for more steps...Step 2.2.1By the Sum Rule, the derivative of with respect to is .Step 2.2.2Evaluate .Tap for more steps...Step 2.2.2.1Since is constant with respect to , the derivative of with respect to is .Step 2.2.2.2Rewrite as .Step 2.2.2.3Differentiate using the chain rule, which states that is where and .Tap for more steps...Step 2.2.2.3.1To apply the Chain Rule, set as .Step 2.2.2.3.2Differentiate using the Power Rule which states that is where .Step 2.2.2.3.3Replace all occurrences of with .Step 2.2.2.4Differentiate using the Power Rule which states that is where .Step 2.2.2.5Multiply the exponents in .Tap for more steps...Step 2.2.2.5.1Apply the power rule and multiply exponents, .Step 2.2.2.5.2Combine and .Step 2.2.2.5.3Move the negative in front of the fraction.Step 2.2.2.6To write as a fraction with a common denominator, multiply by .Step 2.2.2.7Combine and .Step 2.2.2.8Combine the numerators over the common denominator.Step 2.2.2.9Simplify the numerator.Tap for more steps...Step 2.2.2.9.1Multiply by .Step 2.2.2.9.2Subtract from .Step 2.2.2.10Move the negative in front of the fraction.Step 2.2.2.11Combine and .Step 2.2.2.12Combine and .Step 2.2.2.13Multiply by by adding the exponents.Tap for more steps...Step 2.2.2.13.1Use the power rule to combine exponents.Step 2.2.2.13.2Combine the numerators over the common denominator.Step 2.2.2.13.3Subtract from .Step 2.2.2.13.4Move the negative in front of the fraction.Step 2.2.2.14Move to the denominator using the negative exponent rule .Step 2.2.2.15Multiply by .Step 2.2.2.16Multiply by .Step 2.2.3Evaluate .Tap for more steps...Step 2.2.3.1Since is constant with respect to , the derivative of with respect to is .Step 2.2.3.2Differentiate using the Power Rule which states that is where .Step 2.2.3.3To write as a fraction with a common denominator, multiply by .Step 2.2.3.4Combine and .Step 2.2.3.5Combine the numerators over the common denominator.Step 2.2.3.6Simplify the numerator.Tap for more steps...Step 2.2.3.6.1Multiply by .Step 2.2.3.6.2Subtract from .Step 2.2.3.7Move the negative in front of the fraction.Step 2.2.3.8Combine and .Step 2.2.3.9Multiply by .Step 2.2.3.10Multiply by .Step 2.2.3.11Multiply by .Step 2.2.3.12Move to the denominator using the negative exponent rule .Step 2.3The second derivative of with respect to is .Step 3Set the second derivative equal to then solve the equation .Tap for more steps...Step 3.1Set the second derivative equal to .Step 3.2Find the LCD of the terms in the equation.Tap for more steps...Step 3.2.1Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.Step 3.2.2Since contains 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 .Step 3.2.3The 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.Step 3.2.4 has factors of and .Step 3.2.5The number is not a prime number because it only has one positive factor, which is itself.Not primeStep 3.2.6The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.Step 3.2.7Multiply by .Step 3.2.8The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.Step 3.2.9The LCM for is the numeric part multiplied by the variable part.Step 3.3Multiply each term in by to eliminate the fractions.Tap for more steps...Step 3.3.1Multiply each term in by .Step 3.3.2Simplify the left side.Tap for more steps...Step 3.3.2.1Simplify each term.Tap for more steps...Step 3.3.2.1.1Cancel the common factor of .Tap for more steps...Step 3.3.2.1.1.1Move the leading negative in into the numerator.Step 3.3.2.1.1.2Cancel the common factor.Step 3.3.2.1.1.3Rewrite the expression.Step 3.3.2.1.2Cancel the common factor of .Tap for more steps...Step 3.3.2.1.2.1Move the leading negative in into the numerator.Step 3.3.2.1.2.2Factor out of .Step 3.3.2.1.2.3Factor out of .Step 3.3.2.1.2.4Cancel the common factor.Step 3.3.2.1.2.5Rewrite the expression.Step 3.3.2.1.3Divide by .Step 3.3.2.1.4Simplify.Step 3.3.3Simplify the right side.Tap for more steps...Step 3.3.3.1Multiply .Tap for more steps...Step 3.3.3.1.1Multiply by .Step 3.3.3.1.2Multiply by .Step 3.4Solve the equation.Tap for more steps...Step 3.4.1Add to both sides of the equation.Step 3.4.2Divide each term in by and simplify.Tap for more steps...Step 3.4.2.1Divide each term in by .Step 3.4.2.2Simplify the left side.Tap for more steps...Step 3.4.2.2.1Cancel the common factor of .Tap for more steps...Step 3.4.2.2.1.1Cancel the common factor.Step 3.4.2.2.1.2Divide by .Step 3.4.2.3Simplify the right side.Tap for more steps...Step 3.4.2.3.1Cancel the common factor of and .Tap for more steps...Step 3.4.2.3.1.1Factor out of .Step 3.4.2.3.1.2Cancel the common factors.Tap for more steps...Step 3.4.2.3.1.2.1Factor out of .Step 3.4.2.3.1.2.2Cancel the common factor.Step 3.4.2.3.1.2.3Rewrite the expression.Step 3.4.2.3.2Move the negative in front of the fraction.Step 4Find the points where the second derivative is .Tap for more steps...Step 4.1Substitute in to find the value of .Tap for more steps...Step 4.1.1Replace the variable with in the expression.Step 4.1.2Simplify the result.Tap for more steps...Step 4.1.2.1Simplify each term.Tap for more steps...Step 4.1.2.1.1Use the power rule to distribute the exponent.Tap for more steps...Step 4.1.2.1.1.1Apply the product rule to .Step 4.1.2.1.1.2Apply the product rule to .Step 4.1.2.1.2Rewrite as .Step 4.1.2.1.3Apply the power rule and multiply exponents, .Step 4.1.2.1.4Cancel the common factor of .Tap for more steps...Step 4.1.2.1.4.1Cancel the common factor.Step 4.1.2.1.4.2Rewrite the expression.Step 4.1.2.1.5Raise to the power of .Step 4.1.2.1.6Multiply by .Step 4.1.2.1.7One to any power is one.Step 4.1.2.1.8Combine and .Step 4.1.2.1.9Use the power rule to distribute the exponent.Tap for more steps...Step 4.1.2.1.9.1Apply the product rule to .Step 4.1.2.1.9.2Apply the product rule to .Step 4.1.2.1.10Rewrite as .Step 4.1.2.1.11Apply the power rule and multiply exponents, .Step 4.1.2.1.12Cancel the common factor of .Tap for more steps...Step 4.1.2.1.12.1Cancel the common factor.Step 4.1.2.1.12.2Rewrite the expression.Step 4.1.2.1.13Raise to the power of .Step 4.1.2.1.14One to any power is one.Step 4.1.2.1.15Multiply .Tap for more steps...Step 4.1.2.1.15.1Multiply by .Step 4.1.2.1.15.2Combine and .Step 4.1.2.1.16Move to the denominator using the negative exponent rule .Step 4.1.2.1.17Multiply by by adding the exponents.Tap for more steps...Step 4.1.2.1.17.1Use the power rule to combine exponents.Step 4.1.2.1.17.2To write as a fraction with a common denominator, multiply by .Step 4.1.2.1.17.3Combine and .Step 4.1.2.1.17.4Combine the numerators over the common denominator.Step 4.1.2.1.17.5Simplify the numerator.Tap for more steps...Step 4.1.2.1.17.5.1Multiply by .Step 4.1.2.1.17.5.2Subtract from .Step 4.1.2.2Combine fractions.Tap for more steps...Step 4.1.2.2.1Combine the numerators over the common denominator.Step 4.1.2.2.2Add and .Step 4.1.2.3The final answer is .Step 4.2The point found by substituting in is . This point can be an inflection point.Step 5Split into intervals around the points that could potentially be inflection points.Step 6Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.Tap for more steps...Step 6.1Replace the variable with in the expression.Step 6.2Simplify the result.Tap for more steps...Step 6.2.1To write as a fraction with a common denominator, multiply by .Step 6.2.2Write each expression with a common denominator of , by multiplying each by an appropriate factor of .Tap for more steps...Step 6.2.2.1Multiply by .Step 6.2.2.2Multiply by by adding the exponents.Tap for more steps...Step 6.2.2.2.1Move .Step 6.2.2.2.2Use the power rule to combine exponents.Step 6.2.2.2.3Combine the numerators over the common denominator.Step 6.2.2.2.4Add and .Step 6.2.3Combine the numerators over the common denominator.Step 6.2.4Simplify the numerator.Tap for more steps...Step 6.2.4.1Divide by .Step 6.2.4.2Raise to the power of .Step 6.2.4.3Multiply by .Step 6.2.4.4Add and .Step 6.2.5Rewrite as .Step 6.2.6Cancel the common factors.Tap for more steps...Step 6.2.6.1Factor out of .Step 6.2.6.2Cancel the common factor.Step 6.2.6.3Rewrite the expression.Step 6.2.7The final answer is .Step 6.3At , the second derivative is . Since this is positive, the second derivative is increasing on the interval .Increasing on since Increasing on since Step 7Substitute a value from the interval into the second derivative to determine if it is increasing or decreasing.Tap for more steps...Step 7.1Replace the variable with in the expression.Step 7.2Simplify the result.Tap for more steps...Step 7.2.1To write as a fraction with a common denominator, multiply by .Step 7.2.2Write each expression with a common denominator of , by multiplying each by an appropriate factor of .Tap for more steps...Step 7.2.2.1Multiply by .Step 7.2.2.2Multiply by by adding the exponents.Tap for more steps...Step 7.2.2.2.1Move .Step 7.2.2.2.2Use the power rule to combine exponents.Step 7.2.2.2.3Combine the numerators over the common denominator.Step 7.2.2.2.4Add and .Step 7.2.3Combine the numerators over the common denominator.Step 7.2.4Simplify the numerator.Tap for more steps...Step 7.2.4.1Divide by .Step 7.2.4.2Raise to the power of .Step 7.2.4.3Multiply by .Step 7.2.4.4Add and .Step 7.2.5Rewrite as .Step 7.2.6Cancel the common factors.Tap for more steps...Step 7.2.6.1Factor out of .Step 7.2.6.2Cancel the common factor.Step 7.2.6.3Rewrite the expression.Step 7.2.7Move the negative in front of the fraction.Step 7.2.8The final answer is .Step 7.3At , the second derivative is . Since this is negative, the second derivative is decreasing on the interval Decreasing on since Decreasing on since Step 8An 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 9

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