Find the Local Maxima and Minima f(x)=8x-x^2

Problem

ƒ(x)=8*x−x2

Solution

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

d(ƒ(x))/d(x)=8−2*x

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

8−2*x=0

−2*x=−8

x=4

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

d2(ƒ(x))/(d(x)2)=−2

4. Evaluate the second derivative at the critical point x=4 Since the second derivative is a constant −2 which is less than zero, the function is concave down at this point.

ƒ(4)″=−2

−2<0⇒local maximum at *x=4

5. Calculate the y-coordinate of the local maximum by substituting x=4 back into the original function.

ƒ(4)=8*(4)−(4)2

ƒ(4)=32−16

ƒ(4)=16

Final Answer

Local Maximum: *(4,16), Local Minimum: None

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