Q87. Let the maximum and minimum values of $\left( \sqrt { 8 x - x ^ { 2 } - 12 } - 4 \right) ^ { 2 } + ( x - 7 ) ^ { 2 } , x \in \mathbf { R }$ be M and m , respectively. Then $\mathrm { M } ^ { 2 } - \mathrm { m } ^ { 2 }$ is equal to $\_\_\_\_$

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6 .$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leqq x \leqq \mathbf { G } .$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leq x \leqq \mathbf { G }$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

We are to find the range of the values of a real number $t$ such that the maximum value of the cubic function $$f(x) = \frac{1}{3}x^3 - \frac{t+2}{2}x^2 + 2tx + \frac{2}{3}$$ over the interval $x \leqq 4$ is greater than 6.
First of all, since the derivative of $f(x)$ is $$f'(x) = (x - \mathbf{A})(x - t),$$ we consider the problem by dividing the range of the values of $t$ as follows:
(i) When $t > \mathbf{A}$, $f(x)$ has a local maximum at $x = \mathbf{A}$ and a local minimum at $x = t$. Since $f(4) = \mathbf{B}$, we only have to find the range of the values of $t$ satisfying $f(\mathbf{A}) > 6$.
(ii) When $t = \mathbf{A}$, the maximum value of $f(x)$ over the interval $x \leqq 4$ is $f(\mathbf{C}) = \mathbf{D}$, and hence the condition is not satisfied.
(iii) When $t < \mathbf{A}$, $f(x)$ has a local maximum at $x = t$ and a local minimum at $x = \mathbf{A}$. Since $f(4) = \mathbf{B}$, we only have to find the range of the values of $t$ satisfying $f(t) > 6$.
Here, we note $$f(t) - 6 = -\frac{1}{6}(t + \mathbf{E})(t - \mathbf{EF})^2.$$
From the above, the range of the values of $t$ is to be determined.

Q2 Consider the quadratic function
$$f ( x ) = \frac { 3 } { 4 } x ^ { 2 } - 3 x + 4$$
Let $a$ and $b$ be real numbers satisfying $0 < a < b$ and $2 < b$. We are to find the values of $a$ and $b$ such that the range of the values of the function $y = f ( x )$ on $a \leqq x \leqq b$ is $a \leqq y \leqq b$.
Since the equation of the axis of symmetry of the graph of $y = f ( x )$ is $x = \mathbf { M }$, we divide the problem into two cases as follows:
(i) $\mathbf{M} \leqq a$;
(ii) $0 < a < \mathbf{M}$.
In the case of (i), since the values of $f ( x )$ increase with $x$ on $a \leqq x \leqq b$, the equations $f ( a ) = a$ and $f ( b ) = b$ have to be satisfied. By solving these, we obtain $a = \frac { \mathbf { N } } { \mathbf { O } }$ and $b = \mathbf { P }$. However, this $a$ does not satisfy (i).
In the case of (ii), since the minimum value of $f ( x )$ on $a \leqq x \leqq b$ is $\mathbf { Q }$, we have
$$a = \mathbf { R } .$$
This satisfies (ii). Then since $f ( a ) = \frac { \mathbf { S } } { \mathbf { T } } < b$, we have $f ( b ) = b$. Hence, we obtain
$$b = \mathbf { U } .$$

Let $a$ be a real number satisfying $a \geqq 0$. We are to express the maximum value $M$ of the function $f(x) = |x^2 - 2x|$ on the range $a \leqq x \leqq a + 1$ in terms of $a$. Furthermore, we are to find the minimum value of $M$ over the range $a \geqq 0$.
(1) The function $f(x)$ can be expressed without using the absolute value symbol as follows: when $x \leqq \mathbf{M}$ or $x \geqq \mathbf{N}$, then $f(x) = x^2 - 2x$; when $\mathbf{M} < x < \mathbf{N}$, then $f(x) = -x^2 + 2x$.
Hence, the maximum value of $f(x)$ on $a \leqq x \leqq a + 1$ is the following: when $0 \leqq a \leqq \mathbf{O}$, then $M = \mathbf{P}$; when $\mathbf{O} < a \leqq \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = -a^2 + \frac{\mathbf{T}}{\mathbf{S}}a$; when $a > \frac{\mathbf{Q} + \sqrt{\mathbf{R}}}{\mathbf{S}}$, then $M = a^2 - \mathbf{U}$.
(2) The minimum value of $M$ over the range $a \geqq 0$ is $\frac{\sqrt{\mathbf{V}}}{\mathbf{W}}$.

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science applicants should turn to page 14. In this question we shall consider the function $f ( x )$ defined by
$$f ( x ) = x ^ { 2 } - 2 p x + 3$$
where $p$ is a constant.
(i) Show that the function $f ( x )$ has one stationary value in the range $0 < x < 1$ if $0 < p < 1$, and no stationary values in that range otherwise.
In the remainder of the question we shall be interested in the smallest value attained by $f ( x )$ in the range $0 \leqslant x \leqslant 1$. Of course, this value, which we shall call $m$, will depend on $p$.
(ii) Show that if $p \geqslant 1$ then $m = 4 - 2 p$.
(iii) What is the value of $m$ if $p \leqslant 0$ ?
(iv) Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.
(v)Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $- 2 \leqslant p \leqslant 2$. [Figure]

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science applicants should turn to page 14. In this question we shall consider the function $f ( x )$ defined by
$$f ( x ) = x ^ { 2 } - 2 p x + 3$$
where $p$ is a constant.
(i) Show that the function $f ( x )$ has one stationary value in the range $0 < x < 1$ if $0 < p < 1$, and no stationary values in that range otherwise.
In the remainder of the question we shall be interested in the smallest value attained by $f ( x )$ in the range $0 \leqslant x \leqslant 1$. Of course, this value, which we shall call $m$, will depend on $p$.
(ii) Show that if $p \geqslant 1$ then $m = 4 - 2 p$.
(iii) What is the value of $m$ if $p \leqslant 0$ ?
(iv) Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.
(v)Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $- 2 \leqslant p \leqslant 2$. [Figure]

4. In this question we shall consider the function $f ( x )$ defined by
$$f ( x ) = x ^ { 2 } - 2 p x + 3$$
where $p$ is a constant.
(i) Show that the function $f ( x )$ has one stationary value in the range $0 < x < 1$ if $0 < p < 1$, and no stationary values in that range otherwise.
In the remainder of the question, we shall be interested in the smallest value attained by $f ( x )$ in the range $0 \leq x \leq 1$. Of course, this value, which we shall call $m$, will depend on $p$.
(ii) Show that if $p \geq 1$ then $m = 4 - 2 p$.
(iii) What is the value of $m$ if $p \leq 0$ ?
(iv) Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.
(v) Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $- 2 \leq p \leq 2$. [Figure]
5. [Figure]
The diagram represents an array of $N$ electric lights arranged in a circle. Initially, each light may be set to be ON or OFF in an arbitrary way. After one second the settings are updated according to the following rule which determine the new state of a bulb in terms of the initial states of that bulb and the one just next to it in the clockwise direction. if initially bulb $n$ and bulb $n + 1$ are in the same state (i.e. either both OFF or both ON) then after 1 second bulb $n$ will be OFF; if initially bulb $n$ and bulb $n + 1$ are in different states (i.e. one OFF the other ON) then after 1 second bulb $n$ will be ON ; Of course, if $n = N$, we replace $n + 1$ with 1 in the above. Subsequently, the settings are updated each second by reapplying the same rule.
(i) Explain why after one second there cannot be exactly one bulb ON .
(ii) More generally, explain why after one second there cannot be an odd number of bulbs ON.
(iii) Show that the state of bulb $n$ after 2 seconds is completely determined by the initial states of bulbs $n$ and $n + 2$ (with appropriate modifications when $n = N$ or $n = N - 1 )$.
(iv) The initial states of which bulbs determine the state of bulb $n$ after 4 seconds?
(v) Show that if $N = 8$ then, irrespective of the initial settings, all bulbs will eventually be OFF. How long will this take?

Given the function $f ( x ) = \sqrt [ 3 ] { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$, find:\ a) ( 0.25 points) Study whether it is even or odd.\ b) ( 0.75 points) Study its differentiability at the point $x = 1$.\ c) (1.5 points) Study its relative and absolute extrema.

A team of engineers conducts fuel consumption tests for a new hybrid vehicle. The fuel consumption in liters per 100 kilometers as a function of speed, measured in tens of kilometers per hour, is
$$c ( v ) = \left\{ \begin{array} { l l l } \frac { 5 v } { 3 } & \text { if } & 0 \leq v < 3 \\ 14 - 4 v + \frac { v ^ { 2 } } { 3 } & \text { if } & v \geq 3 \end{array} \right.$$
a) (1 point) If in a first test the vehicle must travel at more than 3 tens of kilometers per hour, at what speed should the vehicle travel to obtain minimum fuel consumption?\ b) (1.5 points) If in another test the vehicle must travel at a speed $v$ such that $1 \leq v \leq 8$, what will be the maximum and minimum possible fuel consumption of the vehicle?

3. The temperature function for a certain desert region during a certain period is $f(t) = -t^2 + 10t + 11$, where $1 \leq t \leq 10$. The maximum temperature difference in this region during this period is
(1) 9
(2) 16
(3) 20
(4) 25
(5) 36

Find the maximum value of
$$4^{\sin x} - 4 \times 2^{\sin x} + \frac{17}{4}$$
for real $x$.

Given that
$$\left( a ^ { 3 } + \frac { 2 } { b ^ { 3 } } \right) \left( \frac { 2 } { a ^ { 3 } } - b ^ { 3 } \right) = \sqrt { 2 }$$
where $a$ and $b$ are real numbers, what is the least value of $a b$ ?

The diagram shows the graph of $y = f ( x )$
The function $f$ attains its maximum value of 2 at $x = 1$, and its minimum value of - 2 at $x = - 1$
Find the difference between the maximum and minimum values of $( f ( x ) ) ^ { 2 } - f ( x )$

5

Let $\alpha$ be a positive real number. Define the function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ as the square of the distance AP between the two points $\mathrm{A}(-\alpha,\ -3)$ and $\mathrm{P}(\theta + \sin\theta,\ \cos\theta)$ in the coordinate plane.

1. [(1)] Show that there exists exactly one $\theta$ in the range $0 < \theta < \pi$ such that $f'(\theta) = 0$.
   
2. [(2)] Find the range of $\alpha$ such that the following holds: [6pt] The function $f(\theta)$ of $\theta$ for $0 \leq \theta \leq \pi$ attains its maximum at some point in the interval $0 < \theta < \dfrac{\pi}{2}$.

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1

Consider the following function $f(x)$.
$$f(x) = (\cos x)\log(\cos x) - \cos x + \int_0^x (\cos t)\log(\cos t)\, dt \quad \left(0 \leq x < \frac{\pi}{2}\right)$$

• [(1)] Show that $f(x)$ has a minimum value on the interval $0 \leq x < \dfrac{\pi}{2}$.
• [(2)] Find the minimum value of $f(x)$ on the interval $0 \leq x < \dfrac{\pi}{2}$.

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2

Consider the following function $f(x)$. $\displaystyle f(x) = \int_0^1 \frac{|t-x|}{1+t^2}\,dt \quad (0 \leq x \leq 1)$

(1) Find all real numbers $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan\alpha) = 0$.

(2) For the value of $\alpha$ found in (1), find the value of $\tan\alpha$.

(3) Find the maximum value and the minimum value of $f(x)$ on the interval $0 \leq x \leq 1$. If necessary, you may use the fact that $0.69 < \log 2 < 0.7$.
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$$f(x) = x^{4} - 5x^{2} + 4$$
What is the maximum value of the function on the interval $\left[\frac{-1}{2}, \frac{1}{2}\right]$?
A) 8
B) 6
C) 4
D) 2
E) 0