Find The Local Maxima And Minima F(x)=2x^6-3x^4 | Mathway

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Calculus Examples Popular Problems Calculus Find the Local Maxima and Minima f(x)=2x^6-3x^4 Step 1Find the first derivative of the function.Tap for more steps...Step 1.1By the Sum Rule, the derivative of with respect to is .Step 1.2Evaluate .Tap for more steps...Step 1.2.1Since is constant with respect to , the derivative of with respect to is .Step 1.2.2Differentiate using the Power Rule which states that is where .Step 1.2.3Multiply by .Step 1.3Evaluate .Tap for more steps...Step 1.3.1Since is constant with respect to , the derivative of with respect to is .Step 1.3.2Differentiate using the Power Rule which states that is where .Step 1.3.3Multiply by .Step 2Find the second derivative of the function.Tap for more steps...Step 2.1By the Sum Rule, the derivative of with respect to is .Step 2.2Evaluate .Tap for more steps...Step 2.2.1Since is constant with respect to , the derivative of with respect to is .Step 2.2.2Differentiate using the Power Rule which states that is where .Step 2.2.3Multiply by .Step 2.3Evaluate .Tap for more steps...Step 2.3.1Since is constant with respect to , the derivative of with respect to is .Step 2.3.2Differentiate using the Power Rule which states that is where .Step 2.3.3Multiply by .Step 3To find the local maximum and minimum values of the function, set the derivative equal to and solve.Step 4Find the first derivative.Tap for more steps...Step 4.1Find the first derivative.Tap for more steps...Step 4.1.1By the Sum Rule, the derivative of with respect to is .Step 4.1.2Evaluate .Tap for more steps...Step 4.1.2.1Since is constant with respect to , the derivative of with respect to is .Step 4.1.2.2Differentiate using the Power Rule which states that is where .Step 4.1.2.3Multiply by .Step 4.1.3Evaluate .Tap for more steps...Step 4.1.3.1Since is constant with respect to , the derivative of with respect to is .Step 4.1.3.2Differentiate using the Power Rule which states that is where .Step 4.1.3.3Multiply by .Step 4.2The first derivative of with respect to is .Step 5Set the first derivative equal to then solve the equation .Tap for more steps...Step 5.1Set the first derivative equal to .Step 5.2Factor the left side of the equation.Tap for more steps...Step 5.2.1Factor out of .Tap for more steps...Step 5.2.1.1Factor out of .Step 5.2.1.2Factor out of .Step 5.2.1.3Factor out of .Step 5.2.2Rewrite as .Step 5.2.3Factor.Tap for more steps...Step 5.2.3.1Since both terms are perfect squares, factor using the difference of squares formula, where and .Step 5.2.3.2Remove unnecessary parentheses.Step 5.3If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .Step 5.4Set equal to and solve for .Tap for more steps...Step 5.4.1Set equal to .Step 5.4.2Solve for .Tap for more steps...Step 5.4.2.1Take the specified root of both sides of the equation to eliminate the exponent on the left side.Step 5.4.2.2Simplify .Tap for more steps...Step 5.4.2.2.1Rewrite as .Step 5.4.2.2.2Pull terms out from under the radical, assuming real numbers.Step 5.5Set equal to and solve for .Tap for more steps...Step 5.5.1Set equal to .Step 5.5.2Subtract from both sides of the equation.Step 5.6Set equal to and solve for .Tap for more steps...Step 5.6.1Set equal to .Step 5.6.2Add to both sides of the equation.Step 5.7The final solution is all the values that make true.Step 6Find the values where the derivative is undefined.Tap for more steps...Step 6.1The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.Step 7Critical points to evaluate.Step 8Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.Step 9Evaluate the second derivative.Tap for more steps...Step 9.1Simplify each term.Tap for more steps...Step 9.1.1Raising to any positive power yields .Step 9.1.2Multiply by .Step 9.1.3Raising to any positive power yields .Step 9.1.4Multiply by .Step 9.2Add and .Step 10Since there is at least one point with or undefined second derivative, apply the first derivative test.Tap for more steps...Step 10.1Split into separate intervals around the values that make the first derivative or undefined.Step 10.2Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.Tap for more steps...Step 10.2.1Replace the variable with in the expression.Step 10.2.2Simplify the result.Tap for more steps...Step 10.2.2.1Simplify each term.Tap for more steps...Step 10.2.2.1.1Raise to the power of .Step 10.2.2.1.2Multiply by .Step 10.2.2.1.3Raise to the power of .Step 10.2.2.1.4Multiply by .Step 10.2.2.2Add and .Step 10.2.2.3The final answer is .Step 10.3Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.Tap for more steps...Step 10.3.1Replace the variable with in the expression.Step 10.3.2Simplify the result.Tap for more steps...Step 10.3.2.1Simplify each term.Tap for more steps...Step 10.3.2.1.1Raise to the power of .Step 10.3.2.1.2Multiply by .Step 10.3.2.1.3Raise to the power of .Step 10.3.2.1.4Multiply by .Step 10.3.2.2Add and .Step 10.3.2.3The final answer is .Step 10.4Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.Tap for more steps...Step 10.4.1Replace the variable with in the expression.Step 10.4.2Simplify the result.Tap for more steps...Step 10.4.2.1Simplify each term.Tap for more steps...Step 10.4.2.1.1Raise to the power of .Step 10.4.2.1.2Multiply by .Step 10.4.2.1.3Raise to the power of .Step 10.4.2.1.4Multiply by .Step 10.4.2.2Subtract from .Step 10.4.2.3The final answer is .Step 10.5Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.Tap for more steps...Step 10.5.1Replace the variable with in the expression.Step 10.5.2Simplify the result.Tap for more steps...Step 10.5.2.1Simplify each term.Tap for more steps...Step 10.5.2.1.1Raise to the power of .Step 10.5.2.1.2Multiply by .Step 10.5.2.1.3Raise to the power of .Step 10.5.2.1.4Multiply by .Step 10.5.2.2Subtract from .Step 10.5.2.3The final answer is .Step 10.6Since the first derivative changed signs from negative to positive around , then is a local minimum. is a local minimumStep 10.7Since the first derivative changed signs from positive to negative around , then is a local maximum. is a local maximumStep 10.8Since the first derivative changed signs from negative to positive around , then is a local minimum. is a local minimumStep 10.9These are the local extrema for . is a local minimum is a local maximum is a local minimum is a local minimum is a local maximum is a local minimumStep 11

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