Find the Local Maxima and Minima x^3-3x

Problem

x3−3*x

Solution

1. Find the first derivative of the function to identify critical points.

(d(x3)−3*x)/d(x)=3*x2−3

2. Set the derivative to zero and solve for x to find the critical values.

3*x2−3=0

3*(x2−1)=0

x2=1

x=1,x=−1

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

(d(3)*x2−3)/d(x)=6*x

4. Evaluate the second derivative at x=−1

6*(−1)=−6

Since −6<0 the function is concave down, indicating a local maximum at x=−1

5. Evaluate the second derivative at x=1

6*(1)=6

Since 6>0 the function is concave up, indicating a local minimum at x=1

6. Calculate the y-coordinates by substituting the critical values back into the original function.

(−1)3−3*(−1)=−1+3=2

(1)3−3*(1)=1−3=−2

Final Answer

Local Maxima: *(−1,2), Local Minima: *(1,−2)

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