Families of Functions and Transformations - Questions
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Consider the family of functions of the form \(f(x) = {(x - 1)^2}{(x - b)}\), where \(b>0\). Find the value of \(b\) if stationary points occur when \(x=3\)
\begin{align}
&f(x)=(x-1)^{2}(x-b) \\
&\begin{aligned}
\text{Let } u &=(x-1)^{2} \quad\;\; v=x-b \\
du &=2(x-1) \quad dv=1 \\
f^{\prime}(x) &=(x-1)^{2}+2(x-1)(x-b) \\
&=(x-1)(x-1+2(x-b)) \\
&=(x-1)(x-1+2 x-2 b) \\
&=(x-1)(3 x-2 b-1)
\end{aligned}\\
&\text{Let } f^{\prime}(x)=0\\
&(x-1)(3 x-2 b-1)=0 \\
&\begin{aligned}
x=1 \;\text { or }\; 3x-2b-1&=0\\
2 b &=3 x-1 \\
b &=\frac{3 x-1}{2} \\
\text {wh}&\text{en } x=3 \\
b &=\frac{9-1}{2} \\
\therefore b &=4
\end{aligned}
\end{align}
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