determine the x - value of all critical points for the following function.\n f(x)=\frac{3 x^{2}}{x - 4}…

determine the x - value of all critical points for the following function.\n f(x)=\frac{3 x^{2}}{x - 4} \nthe x - value(s) of the critical point(s) of ( f(x)=\frac{3 x^{2}}{x - 4} ) is/are ( x = ) (use a comma to separate answers as needed.)

determine the x - value of all critical points for the following function.\n f(x)=\frac{3 x^{2}}{x - 4} \nthe x - value(s) of the critical point(s) of ( f(x)=\frac{3 x^{2}}{x - 4} ) is/are ( x = ) (use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Find the derivative of ( f(x) )

Use the quotient rule ( \left(\frac{u}{v}\right)'=\frac{u'v - uv'}{v^{2}} ). Here ( u = 3x^{2}), (u'=6x), (v=x - 4), (v' = 1). [ \begin{align*} f'(x)&=\frac{(6x)(x - 4)-3x^{2}(1)}{(x - 4)^{2}}\ &=\frac{6x^{2}-24x-3x^{2}}{(x - 4)^{2}}\ &=\frac{3x^{2}-24x}{(x - 4)^{2}}\ &=\frac{3x(x - 8)}{(x - 4)^{2}} \end{align*} ]

Step2: Set ( f'(x)=0 ) and find ( x )

Set ( 3x(x - 8)=0 ). For ( 3x=0), we get ( x = 0 ). For ( x - 8=0), we get ( x=8 ). Also, check where ( f'(x) ) is undefined. The derivative ( f'(x)=\frac{3x(x - 8)}{(x - 4)^{2}} ) is undefined when ( x = 4 ), but ( x = 4 ) is not in the domain of ( f(x) ) (since ( f(4)=\frac{3\times4^{2}}{4 - 4}) is undefined).

Answer:

(0,8)