Find the Local Maxima and Minima f(x)=3x^3-9x

Problem

ƒ(x)=3*x3−9*x

Solution

1. Find the first derivative of the function to determine the slope of the tangent line.

d(ƒ(x))/d(x)=9*x2−9

2. Set the derivative to zero to find the critical points where the slope is horizontal.

9*x2−9=0

3. Solve for x by factoring or using algebraic manipulation.

9*(x2−1)=0

x2=1

x=1,x=−1

4. Find the second derivative to apply the Second Derivative Test for concavity.

d2(ƒ(x))/(d(x)2)=18*x

5. Evaluate the second derivative at the critical points to determine if they are maxima or minima.

At *x=1:18*(1)=18>0⇒Local Minimum

At *x=−1:18*(−1)=−18<0⇒Local Maximum

6. Calculate the y-coordinates by substituting the critical points back into the original function ƒ(x)

ƒ(1)=3*(1)3−9*(1)=−6

ƒ*(−1)=3*(−1)3−9*(−1)=6

Final Answer

Local Maximum: *(−1,6), Local Minimum: *(1,−6)

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