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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 \# \( 2 \square \)

Ask by Powers Cross. in the United States
Mar 17,2025

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**a) \( f(x)=x^3-6x^2+9x \)** - Blank \# \(1 \square\) \(x=1\) - Blank \# \(2 \square\) \(x=3\) **b) \( f(x)=\frac{x^2}{x-2} \)** - Blank \# \(1 \square\) \(x=0\) - Blank \# \(2 \square\) \(x=4\)

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Beyond the Answer

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: \( x= 3 \)

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