Find The Local Maxima And Minima G(x)=x^3-3x | Mathway

Enter a problem... Calculus Examples Popular Problems Calculus Find the Local Maxima and Minima g(x)=x^3-3x Step 1Find the first derivative of the function.Tap for more steps...Step 1.1Differentiate.Tap for more steps...Step 1.1.1By the Sum Rule, the derivative of with respect to is .Step 1.1.2Differentiate using the Power Rule which states that is where .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 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 2.3Differentiate using the Constant Rule.Tap for more steps...Step 2.3.1Since is constant with respect to , the derivative of with respect to is .Step 2.3.2Add and .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 4.1.1Differentiate.Tap for more steps...Step 4.1.1.1By the Sum Rule, the derivative of with respect to is .Step 4.1.1.2Differentiate using the Power Rule which states that is where .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 4.2The 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 5.2Add to both sides of the equation.Step 5.3Divide each term in by and simplify.Tap for more steps...Step 5.3.1Divide each term in by .Step 5.3.2Simplify the left side.Tap for more steps...Step 5.3.2.1Cancel the common factor of .Tap for more steps...Step 5.3.2.1.1Cancel the common factor.Step 5.3.2.1.2Divide by .Step 5.3.3Simplify the right side.Tap for more steps...Step 5.3.3.1Divide by .Step 5.4Take the specified root of both sides of the equation to eliminate the exponent on the left side.Step 5.5Any root of is .Step 5.6The complete solution is the result of both the positive and negative portions of the solution.Tap for more steps...Step 5.6.1First, use the positive value of the to find the first solution.Step 5.6.2Next, use the negative value of the to find the second solution.Step 5.6.3The complete solution is the result of both the positive and negative portions of the solution.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 9Multiply by .Step 10 is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test. is a local minimumStep 11Find the y-value when .Tap for more steps...Step 11.1Replace the variable with in the expression.Step 11.2Simplify the result.Tap for more steps...Step 11.2.1Simplify each term.Tap for more steps...Step 11.2.1.1One to any power is one.Step 11.2.1.2Multiply by .Step 11.2.2Subtract from .Step 11.2.3The final answer is .Step 12Evaluate 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 13Multiply by .Step 14 is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test. is a local maximumStep 15Find the y-value when .Tap for more steps...Step 15.1Replace the variable with in the expression.Step 15.2Simplify the result.Tap for more steps...Step 15.2.1Simplify each term.Tap for more steps...Step 15.2.1.1Raise to the power of .Step 15.2.1.2Multiply by .Step 15.2.2Add and .Step 15.2.3The final answer is .Step 16These are the local extrema for . is a local minima is a local maximaStep 17

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