Find the Local Maxima and Minima f(x)=x+49/x

Problem

ƒ(x)=x+49/x

Solution

1. Find the first derivative of the function ƒ(x) by applying the power rule to x and 49*x^(−1)

d(ƒ(x))/d(x)=1−49/(x^2)

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

1−49/(x^2)=0

1=49/(x^2)

x^2=49

x=7,x=−7

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

d^2(ƒ(x))/(d(x)^2)=98/(x^3)

4. Apply the Second Derivative Test by substituting the critical points into ƒ(x)^″

ƒ(7)^″=98/7^3=98/343>0

ƒ^″*(−7)=98/((−7)^3)=−98/343<0

5. Determine the extrema based on the test results: x=7 is a local minimum (concave up) and x=−7 is a local maximum (concave down).

ƒ(7)=7+49/7=14

ƒ*(−7)=−7+49/(−7)=−14

Final Answer

Local Maximum: *(−7,−14), Local Minimum: *(7,14)

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