Find the Local Maxima and Minima f(x)=x^3-27x

Problem

ƒ(x)=x3−27*x

Solution

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

(d(x3)−27*x)/d(x)=3*x2−27

2. Identify critical points by setting the first derivative equal to zero and solving for x

3*x2−27=0

3*x2=27

x2=9

x=±3

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

(d(3)*x2−27)/d(x)=6*x

4. Test the critical point x=3 by substituting it into the second derivative.

ƒ(3)″=6*(3)=18

Since ƒ(3)″>0 the function is concave up, indicating a local minimum at x=3

5. Test the critical point x=−3 by substituting it into the second derivative.

ƒ″*(−3)=6*(−3)=−18

Since ƒ″*(−3)<0 the function is concave down, indicating a local maximum at x=−3

6. Calculate the y-values for the local extrema by substituting the x values back into the original function ƒ(x)

ƒ(3)=(3)3−27*(3)=27−81=−54

ƒ*(−3)=(−3)3−27*(−3)=−27+81=54

Final Answer

Local Maxima: *(−3,54), Local Minima: *(3,−54)

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