Find the Critical Points y=x/(x^2+1)

Problem

y=x/(x2+1)

Solution

1. Identify the condition for critical points by finding where the derivative d(y)/d(x) is equal to zero or undefined.

2. Apply the quotient rule to find the derivative of the function y=u/v where u=x and v=x2+1

3. Calculate the derivative using the formula d(y)/d(x)=(vd(u)/d(x)−ud(v)/d(x))/(v2)

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

4. Simplify the numerator of the derivative.

d(y)/d(x)=(x2+1−2*x2)/((x2+1)2)

d(y)/d(x)=(1−x2)/((x2+1)2)

5. Set the derivative to zero to find the critical points. Since the denominator (x2+1)2 is always positive and never zero, we only need to solve for the numerator.

1−x2=0

6. Solve for x by factoring or taking the square root.

x2=1

x=1,x=−1

Final Answer

x=1,−1

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