Find the Critical Points f(x)=x/(x^2+4)

Problem

ƒ(x)=x/(x2+4)

Solution

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

2. Apply the quotient rule to find the derivative, where u=x and v=x2+4

d(ƒ(x))/d(x)=((x2+4)d(x)/d(x)−xd(x2+4)/d(x))/((x2+4)2)

3. Differentiate the numerator terms.

d(ƒ(x))/d(x)=((x2+4)*(1)−x*(2*x))/((x2+4)2)

4. Simplify the numerator by combining like terms.

d(ƒ(x))/d(x)=(x2+4−2*x2)/((x2+4)2)

d(ƒ(x))/d(x)=(4−x2)/((x2+4)2)

5. Set the derivative to zero to find the critical points. Since the denominator (x2+4)2 is always positive and never zero, we only solve for the numerator.

4−x2=0

6. Solve for x by factoring or using the square root property.

x2=4

x=±2

Final Answer

x=−2,2

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