2.15 Solving inequalities questions for IB Maths AA

Factorize $f(x)$ fully.

Hence solve the inequality $f(x)<0$.

Use your GDC to solve the inequality $2^x>x^2$, for $x>0$.

Use your GDC to solve the inequality $3^x>x^3$, for $x>0$.

Show that, for $x>0$, the inequality $a^x>x^a$ is equivalent to ${\displaystyle \frac{\ln x}{x}}<{\displaystyle \frac{\ln a}{a}}$.

Let $g(x)={\displaystyle \frac{\ln x}{x}}$, for $x>0$.

Find the coordinates of the stationary point on the graph of $y=g(x)$ and show that it is a maximum.

Hence show that ${\mathrm{e}}^x\ge x^{\mathrm{e}}$ for all $x>0$, with equality only when $x=\mathrm{e}$.

Hence determine which is larger, ${\mathrm{e}}^{\pi }$ or ${\pi }^{\mathrm{e}}$, justifying your answer.

You are given that $g(x)<0$ for $0<x<1$, $g(1)=0$ and $g(x)\to 0$ as $x\to \mathrm{\infty }$.

Now let $1<a<\mathrm{e}$. Explain why the equation $g(x)=g(a)$ has exactly two solutions for $x>0$, one of which is $x=a$, and the other of which, $x=b$, satisfies $b>\mathrm{e}$.

Hence write down, in terms of $a$ and $b$, the solution of $a^x>x^a$ for $x>0$, where $1<a<\mathrm{e}$, and state the value of $b$ when $a=2$.

Suggest how the solution of $a^x>x^a$ changes when $a>\mathrm{e}$. Use part (b) to support your answer.

Find the set of values of $a>1$ for which $a^x>x^a$ is true for all $x>a$.

Show that the volume of the box, $V$ cm${}^3$, is given by $V=x(20-2x)(14-2x)$.

Find the values of $x$ for which $V\ge 300$.

Find the maximum volume of the box.

Consider the case $n=2$. Sketch the graph of $y=S_2(x)$ for $0\le x\le 3$, showing the coordinates of the end points and of the points where the gradient changes.

Write down the minimum value of $S_2(x)$ and the values of $x$ for which it occurs.

Consider the case $n=3$. Show that $S_3(x)=4-x$ for $1\le x\le 2$.

Hence or otherwise, find the minimum value of $S_3(x)$ and the value of $x$ at which it occurs.

Let $m_n$ be the minimum value of $S_n(x)$. Use technology to find $m_4$, $m_5$ and $m_6$.

Let $j$ be an integer with $1\le j\le n-1$. Show that, for $j<x<j+1$, the gradient of the graph of $y=S_n(x)$ is $2j-n$.

Hence, explain why the minimum value of $S_n$ occurs only at $x={\displaystyle \frac{n+1}{2}}$ when $n$ is odd, and for every $x$ with ${\displaystyle \frac{n}{2}}\le x\le {\displaystyle \frac{n}{2}}+1$ when $n$ is even.

Let $n=2p$, where $p\in {\mathbb{Z}}^{+}$. Show that $m_n=p^2$.

Let $n=2p+1$, where $p\in {\mathbb{Z}}^{+}$. Find $m_n$ in terms of $p$, and hence show that $m_n={\displaystyle \frac{n^2-1}{4}}$.

Five houses stand beside a straight road, at distances $0$, $2$, $3$, $7$ and $11$ km from a junction. A delivery depot is to be built beside the road, $x$ km from the junction, so that the total distance from the depot to the five houses is as small as possible.

By considering a function similar to $S_n$, find the value of $x$ and the minimum total distance.

A sixth house is built at a distance of $12$ km from the junction. Find the possible positions of the depot that minimise the total distance to the six houses.