Find Where Increasing/Decreasing F(x)=1/3x^3-3/2x^2-4x+2 | Mathway

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Calculus Examples Popular Problems Calculus Find Where Increasing/Decreasing Using Derivatives f(x)=1/3x^3-3/2x^2-4x+2 Step 1Find the first derivative.Tap for more steps...Step 1.1Find the first derivative.Tap for more steps...Step 1.1.1By the Sum Rule, the derivative of with respect to is .Step 1.1.2Evaluate .Tap for more steps...Step 1.1.2.1Since is constant with respect to , the derivative of with respect to is .Step 1.1.2.2Differentiate using the Power Rule which states that is where .Step 1.1.2.3Combine and .Step 1.1.2.4Combine and .Step 1.1.2.5Cancel the common factor of .Tap for more steps...Step 1.1.2.5.1Cancel the common factor.Step 1.1.2.5.2Divide by .Step 1.1.3Evaluate .Tap for more steps...Step 1.1.3.1Since is constant with respect to , the derivative of with respect to is .Step 1.1.3.2Differentiate using the Power Rule which states that is where .Step 1.1.3.3Multiply by .Step 1.1.3.4Combine and .Step 1.1.3.5Multiply by .Step 1.1.3.6Combine and .Step 1.1.3.7Cancel the common factor of and .Tap for more steps...Step 1.1.3.7.1Factor out of .Step 1.1.3.7.2Cancel the common factors.Tap for more steps...Step 1.1.3.7.2.1Factor out of .Step 1.1.3.7.2.2Cancel the common factor.Step 1.1.3.7.2.3Rewrite the expression.Step 1.1.3.7.2.4Divide by .Step 1.1.4Evaluate .Tap for more steps...Step 1.1.4.1Since is constant with respect to , the derivative of with respect to is .Step 1.1.4.2Differentiate using the Power Rule which states that is where .Step 1.1.4.3Multiply by .Step 1.1.5Differentiate using the Constant Rule.Tap for more steps...Step 1.1.5.1Since is constant with respect to , the derivative of with respect to is .Step 1.1.5.2Add and .Step 1.2The first derivative of with respect to is .Step 2Set the first derivative equal to then solve the equation .Tap for more steps...Step 2.1Set the first derivative equal to .Step 2.2Factor using the AC method.Tap for more steps...Step 2.2.1Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .Step 2.2.2Write the factored form using these integers.Step 2.3If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .Step 2.4Set equal to and solve for .Tap for more steps...Step 2.4.1Set equal to .Step 2.4.2Add to both sides of the equation.Step 2.5Set equal to and solve for .Tap for more steps...Step 2.5.1Set equal to .Step 2.5.2Subtract from both sides of the equation.Step 2.6The final solution is all the values that make true.Step 3The values which make the derivative equal to are .Step 4Split into separate intervals around the values that make the derivative or undefined.Step 5Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.Tap for more steps...Step 5.1Replace the variable with in the expression.Step 5.2Simplify the result.Tap for more steps...Step 5.2.1Simplify each term.Tap for more steps...Step 5.2.1.1Raise to the power of .Step 5.2.1.2Multiply by .Step 5.2.2Simplify by adding and subtracting.Tap for more steps...Step 5.2.2.1Add and .Step 5.2.2.2Subtract from .Step 5.2.3The final answer is .Step 5.3At the derivative is . Since this is positive, the function is increasing on .Increasing on since Increasing on since Step 6Substitute a value from the interval into the derivative to determine if the function 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 6.2.1Simplify each term.Tap for more steps...Step 6.2.1.1Apply the product rule to .Step 6.2.1.2Raise to the power of .Step 6.2.1.3Raise to the power of .Step 6.2.1.4Multiply .Tap for more steps...Step 6.2.1.4.1Combine and .Step 6.2.1.4.2Multiply by .Step 6.2.1.5Move the negative in front of the fraction.Step 6.2.2Find the common denominator.Tap for more steps...Step 6.2.2.1Multiply by .Step 6.2.2.2Multiply by .Step 6.2.2.3Write as a fraction with denominator .Step 6.2.2.4Multiply by .Step 6.2.2.5Multiply by .Step 6.2.2.6Multiply by .Step 6.2.3Combine the numerators over the common denominator.Step 6.2.4Simplify each term.Tap for more steps...Step 6.2.4.1Multiply by .Step 6.2.4.2Multiply by .Step 6.2.5Simplify the expression.Tap for more steps...Step 6.2.5.1Subtract from .Step 6.2.5.2Subtract from .Step 6.2.5.3Move the negative in front of the fraction.Step 6.2.6The final answer is .Step 6.3At the derivative is . Since this is negative, the function is decreasing on .Decreasing on since Decreasing on since Step 7Substitute a value from the interval into the derivative to determine if the function 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 7.2.1Simplify each term.Tap for more steps...Step 7.2.1.1Raise to the power of .Step 7.2.1.2Multiply by .Step 7.2.2Simplify by subtracting numbers.Tap for more steps...Step 7.2.2.1Subtract from .Step 7.2.2.2Subtract from .Step 7.2.3The final answer is .Step 7.3At the derivative is . Since this is positive, the function is increasing on .Increasing on since Increasing on since Step 8List the intervals on which the function is increasing and decreasing.Increasing on: Decreasing on: Step 9

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