The diagram shows the graph of $y = f(x)$ - HSC - SSCE Mathematics Extension 2 - Question 3 - 2007 - Paper 1

To sketch the graph of $f(|x|)$, first, plot the graph of $f(x)$ for $x eq 0$. Then, reflect the portion of the graph for $x > 0$ across the y-axis to obtain the graph for $x < 0$. Additionally, ensure that the graph meets at the origin, as $f(0)$ will also determine a point on the graph.

To sketch the graph of $f(x) - x$, take the existing graph of $f(x)$ and shift it downwards by the value of $x$. This will create a new graph, where the y-values are decreased by the corresponding x-values.

The given zeros are $α$, $β$, and $γ$. We need a polynomial whose zeros are $2α$, $2β$, and $2γ$. Thus, multiply the original polynomial by $x - 2α$, $x - 2β$, and $x - 2γ$ to yield:

$P(x) = k(x - 2α)(x - 2β)(x - 2γ)$

where k is a constant (commonly taken as 1). To express this in standard cubic form, expand the factors.

For the volume using cylindrical shells, we integrate the radius multiplied by height. The volume $V$ is given by:

$V = 2π imes ext{Radius} imes ext{Height}$

For our region, the volume when rotated about the y-axis is:

$V = 2π imes ext{(height function)} imes ( ext{radius from the y-axis}) dx$

Calculate the definite integral from $x = 1$ to $x = e$.

Resolve the forces:
For horizontal motion:
$N = mg cos θ - mrω^2 sin θ$.
Set up the equations using the weight mg acting downwards and the centripetal force required for circular motion.

$N > 0$ gives: $mg cos θ - mrω^2 sin θ > 0$.
Rearrange to find the conditions on ω, leading to: ω < rac{g cos θ}{r sin θ}.