Find the critical numbers a) \( f(x)=x^{3}-6 x^{2}+9 x \) b) \( f(x)=\frac{x^{2}}{x-2} \) Type \( x= \) lower number,x=higher number Blank \# \( 1 \square \) Blank \# \( 2 \square \)
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To find the critical numbers of the given functions, we need to calculate the derivative and set it equal to zero or find where it is undefined. For a) \( f'(x) = 3x^{2} - 12x + 9 \). Setting this equal to zero gives us the critical points. Factoring or using the quadratic formula will yield \( x = 1 \) and \( x = 3 \). b) \( f(x) = \frac{x^{2}}{x-2} \). First, find the derivative using the quotient rule. The critical numbers will occur where the derivative is zero or undefined. The function has a vertical asymptote at \( x=2 \), and setting the derivative equal to zero will provide additional critical points. Thus, for your request: Blank # 1: \( x= 1 \) Blank # 2: \( x= 3 \)