Find the Critical Points f(x)=(x^2)/(x-6)

Problem

ƒ(x)=(x2)/(x−6)

Solution

1. Identify the domain of the function. The function is undefined when the denominator is zero, which occurs at x=6

2. Apply the quotient rule to find the derivative ƒ(x)′ The quotient rule states 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−6 Then d(u)/d(x)=2*x and d(v)/d(x)=1

4. Substitute these into the quotient rule formula.

ƒ(x)′=((x−6)*(2*x)−(x2)*(1))/((x−6)2)

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

ƒ(x)′=(2*x2−12*x−x2)/((x−6)2)

ƒ(x)′=(x2−12*x)/((x−6)2)

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

x2−12*x=0

7. Factor the quadratic equation to solve for x

x*(x−12)=0

x=0

x=12

8. Check for points where the derivative is undefined. The derivative is undefined at x=6 but since x=6 is not in the domain of the original function, it is not a critical point.

Final Answer

Critical Points: *x=0,x=12

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