Find the Local Maxima and Minima x^2-6x

Problem

x2−6*x

Solution

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

(d(x2)−6*x)/d(x)=2*x−6

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

2*x−6=0

2*x=6

x=3

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

(d(2)*x−6)/d(x)=2

4. Evaluate the second derivative at the critical point x=3

ƒ(3)″=2

5. Determine the nature of the critical point. Since ƒ(3)″>0 the function is concave up at this point, indicating a local minimum. There are no other critical points, so there is no local maximum.

6. Calculate the y-value of the local minimum by substituting x=3 back into the original expression.

3−6*(3)=9−18

9−18=−9

Final Answer

Local Minimum: *(3,−9), Local Maximum: None

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