Find the Critical Points f(x)=(4x)/(x^2+1)

Problem

ƒ(x)=(4*x)/(x2+1)

Solution

1. Identify the domain of the function. Since the denominator x2+1 is always greater than or equal to 1 there are no vertical asymptotes or points where the function is undefined. The domain is all real numbers.

2. Apply the quotient rule to find the derivative ƒ(x)′ The quotient rule is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator and denominator. Let u=4*x and v=x2+1 Then d(u)/d(x)=4 and d(v)/d(x)=2*x

4. Substitute these values into the quotient rule formula.

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

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

d(ƒ(x))/d(x)=(4*x2+4−8*x2)/((x2+1)2)

d(ƒ(x))/d(x)=(4−4*x2)/((x2+1)2)

6. Set the derivative equal to zero to find the critical points. A fraction is zero when its numerator is zero.

4−4*x2=0

7. Solve for x by factoring or isolating the variable.

4*(1−x2)=0

1−x2=0

x2=1

x=±1

Final Answer

x=−1,1

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