Find the Critical Points f(x)=x/(x^2+64)

Problem

ƒ(x)=x/(x^2+64)

Solution

1. Identify the function ƒ(x)=x/(x^2+64) and note that critical points occur where the derivative ƒ(x)^′=0 or where ƒ(x)^′ is undefined.

2. Apply the quotient rule to find the derivative, using the formula d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

3. Calculate the derivative components where u=x and v=x^2+64

d(ƒ(x))/d(x)=((x^2+64)*(1)−(x)*(2*x))/((x^2+64)^2)

4. Simplify the numerator by distributing and combining like terms.

d(ƒ(x))/d(x)=(x^2+64−2*x^2)/((x^2+64)^2)

d(ƒ(x))/d(x)=(64−x^2)/((x^2+64)^2)

5. Set the derivative equal to zero to find the critical values. Since the denominator (x^2+64)^2 is always positive and never zero, we only need to solve for the numerator.

64−x^2=0

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

x^2=64

x=±8

Final Answer

x=−8,8

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