Find the Local Maxima and Minima f(x)=x+9/x

Problem

ƒ(x)=x+9/x

Solution

1. Find the derivative of the function ƒ(x) using the power rule.

d(ƒ(x))/d(x)=1−9/(x2)

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

1−9/(x2)=0

9/(x2)=1

x2=9

x=3,x=−3

3. Find the second derivative to determine the concavity at the critical points.

d2(ƒ(x))/(d(x)2)=18/(x3)

4. Apply the second derivative test for x=3

ƒ(3)″=18/3

ƒ(3)″=18/27

2/3>0

Since the second derivative is positive, a local minimum occurs at x=3

5. Apply the second derivative test for x=−3

ƒ″*(−3)=18/((−3)3)

ƒ″*(−3)=−18/27

−2/3<0

Since the second derivative is negative, a local maximum occurs at x=−3

6. Calculate the function values for the local extrema.

ƒ(3)=3+9/3=6

ƒ*(−3)=−3+9/(−3)=−6

Final Answer

Local Maxima: *(−3,−6), Local Minima: *(3,6)

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