Find the Local Maxima and Minima 3x^2-12x-8

Problem

3*x2−12*x−8

Solution

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

(d(3)*x2−12*x−8)/d(x)=6*x−12

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

6*x−12=0

3. Solve for x to identify the location of the critical point.

6*x=12

x=2

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

(d(6)*x−12)/d(x)=6

5. Evaluate the second derivative at the critical point x=2 Since 6>0 the function is concave up at this point, indicating a local minimum.

ƒ(2)″=6

6. Calculate the y-coordinate by substituting x=2 back into the original expression.

3*(2)2−12*(2)−8=12−24−8

12−24−8=−20

Final Answer

Local Minimum: *(2,−20), Local Maxima: None

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