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

Problem

ƒ(x)=(x2)/(x+1)

Solution

1. Identify the domain of the function to ensure critical points are valid. The function is undefined when the denominator is zero, so x≠−1

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. Calculate the derivatives of the numerator and denominator. Let u=x2 and v=x+1 Then d(u)/d(x)=2*x and d(v)/d(x)=1

4. Substitute into the quotient rule formula:

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

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

d(ƒ(x))/d(x)=(2*x2+2*x−x2)/((x+1)2)

d(ƒ(x))/d(x)=(x2+2*x)/((x+1)2)

6. Set the derivative to zero to find where the slope is horizontal. A fraction is zero when its numerator is zero:

x2+2*x=0

7. Factor the quadratic equation to solve for x

x*(x+2)=0

x=0

x=−2

8. Check for points where the derivative is undefined. The derivative is undefined at x=−1 but since x=−1 is not in the domain of the original function, it is not a critical point.

Final Answer

Critical Points: *x=0,x=−2

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