Find the Local Maxima and Minima y=x/(x^2+25)

Problem

y=x/(x2+25)

Solution

1. Find the first derivative using the quotient rule, where u=x and v=x2+25

d(y)/d(x)=((x2+25)*(1)−(x)*(2*x))/((x2+25)2)

2. Simplify the numerator to find the expression for the derivative.

d(y)/d(x)=(25−x2)/((x2+25)2)

3. Identify critical points by setting the numerator of the derivative equal to zero.

25−x2=0

x2=25

x=5,x=−5

4. Determine the y-coordinates by substituting the critical points back into the original function.

y(5)=5/(5+25)=5/50=1/10

y*(−5)=(−5)/((−5)2+25)=(−5)/50=−1/10

5. Apply the first derivative test to determine the nature of the critical points. For x<−5 d(y)/d(x)<0 for −5<x<5 d(y)/d(x)>0 for x>5 d(y)/d(x)<0

6. Conclude the local extrema based on the sign changes. Since the derivative changes from negative to positive at x=−5 it is a local minimum. Since it changes from positive to negative at x=5 it is a local maximum.

Final Answer

Local Maxima: *(5,1/10), Local Minima: *(−5,−1/10)

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