1998 Paper 3 Q1

Year: 1998
Paper: 3
Question Number: 1

Course: LFM Pure
Section: Differentiation

Difficulty: 1700.0 Banger: 1484.0

differentiation curve sketching stationary points trigonometric rational function chain rule quotient rule inequality

Problem

Let $$ {\rm f}(x)=\sin^2x + 2 \cos x + 1 $$ for $0 \le x \le 2\pi$. Sketch the curve $y={\rm f}(x)$, giving the coordinates of the stationary points. Now let $$ \hspace{0.6in}{\rm g}(x)={a{\rm f}(x)+b \over c{\rm f}(x)+d} \hspace{0.8in} ad\neq bc\,,\; d\neq -3c\,,\; d\neq c\;. $$ Show that the stationary points of $y={\rm g}(x)$ occur at the same values of $x$ as those of $y={\rm f}(x)$, and find the corresponding values of ${\rm g}(x)$. Explain why, if $d/c <-3$ or $d/c>1$, $|{\rm g}(x)|$ cannot be arbitrarily large.

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Difficulty Rating: 1700.0

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Banger Rating: 1484.0

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Problem source

Let
$$
{\rm f}(x)=\sin^2x + 2 \cos x + 1
$$
for $0 \le x \le 2\pi$. Sketch the curve $y={\rm f}(x)$, giving
the coordinates of the stationary points. Now let
$$
\hspace{0.6in}{\rm g}(x)={a{\rm f}(x)+b \over c{\rm f}(x)+d} 
\hspace{0.8in} 
ad\neq bc\,,\; d\neq -3c\,,\;
d\neq c\;.
$$
Show that the stationary points of $y={\rm g}(x)$ occur at the same
values of $x$ as those of $y={\rm f}(x)$, and find the corresponding 
values of ${\rm g}(x)$.
Explain why, if $d/c <-3$ or $d/c>1$, $|{\rm g}(x)|$ 
cannot be arbitrarily large.

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