Find the Local Maxima and Minima f(x)=x+4/x

Problem

ƒ(x)=x+4/x

Solution

1. Find the derivative of the function ƒ(x) with respect to x to determine the slope of the tangent line.

d(ƒ(x))/d(x)=1−4/(x2)

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

1−4/(x2)=0

1=4/(x2)

x2=4

x=2,x=−2

3. Find the second derivative to apply the second derivative test for concavity.

d2(ƒ(x))/(d(x)2)=8/(x3)

4. Test the critical point x=2 by substituting it into the second derivative.

ƒ(2)″=8/2=1

Since ƒ(2)″>0 the function has a local minimum at x=2

ƒ(2)=2+4/2=4

5. Test the critical point x=−2 by substituting it into the second derivative.

ƒ″*(−2)=8/((−2)3)=−1

Since ƒ″*(−2)<0 the function has a local maximum at x=−2

ƒ*(−2)=−2+4/(−2)=−4

Final Answer

Local Maxima: *(−2,−4), Local Minima: *(2,4)

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