Find the Local Maxima and Minima f(x)=x+25/x

Problem

ƒ(x)=x+25/x

Solution

1. Find the first derivative of the function using the power rule.

d(ƒ(x))/d(x)=1−25/(x^2)

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

1−25/(x^2)=0

1=25/(x^2)

x^2=25

x=5,x=−5

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

d^2(ƒ(x))/(d(x)^2)=50/(x^3)

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

ƒ(5)^″=50/5^3

ƒ(5)^″=50/125=0.4

Since ƒ(5)^″>0 a local minimum occurs at x=5

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

ƒ^″*(−5)=50/((−5)^3)

ƒ^″*(−5)=50/(−125)=−0.4

Since ƒ^″*(−5)<0 a local maximum occurs at x=−5

6. Calculate the function values at these points to find the local extrema.

ƒ(5)=5+25/5=10

ƒ*(−5)=−5+25/(−5)=−10

Final Answer

Local Maxima: *(−5,−10), Local Minima: *(5,10)

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