Find the Local Maxima and Minima f(x)=6x-x^2

Problem

ƒ(x)=6*x−x2

Solution

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

d(ƒ(x))/d(x)=6−2*x

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

6−2*x=0

3. Solve for x to identify the location of the potential local extremum.

−2*x=−6

x=3

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

d2(ƒ(x))/(d(x)2)=−2

5. Evaluate the concavity at the critical point x=3 Since the second derivative is −2 (which is less than 0, the function is concave down at this point.

−2<0

6. Determine the y-coordinate by substituting x=3 back into the original function ƒ(x)

ƒ(3)=6*(3)−(3)2

ƒ(3)=18−9

ƒ(3)=9

Final Answer

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

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