5.8 Stationary points, optimisation and inflexion questions for IB Maths AA

Find $f^{\mathrm{\prime }}(x)$.

Find the coordinates of the two stationary points of the graph of $f$.

Find the equation of the tangent to the graph of $f$ at the point where $x=4$.

Express $h$ in terms of $r$.

Let $A$ cm${}^2$ be the area of paper used to make the cup. Show that $A=\sqrt{{\pi }^2{r}^4+{\displaystyle \frac{562500}{r^2}}}$.

Find the value of $r$ that minimises $A$, and the minimum value of $A$.

For the cup with minimum area, find the angle between the slanting side of the cup and its vertical axis.

Each cup is cut from a sector of a circle of paper and the two straight edges are joined without overlap. For the cup with minimum area, find the angle of the sector, in degrees.

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 $h={\displaystyle \frac{45}{r^2}}-{\displaystyle \frac{2r}{3}}$.

The external surface of the tank, including the base, is $S$ m${}^2$. Show that $S={\displaystyle \frac{5}{3}}\pi r^2+{\displaystyle \frac{90\pi }{r}}$.

Find ${\displaystyle \frac{\mathrm{d}S}{\mathrm{d}r}}$.

Find the value of $r$ for which $S$ has a minimum value.

Use the second derivative to justify that this value of $r$ gives a minimum value of $S$.

Find the minimum value of $S$, and the corresponding height of the cylinder.

Consider the case $\theta =\pi$. Find the value of $r$.

Hence, find the volume of the cone when $\theta =\pi$.

Show that $r={\displaystyle \frac{5\theta }{\pi }}$.

Show that $V={\displaystyle \frac{25{\theta }^2}{3\pi }}\sqrt{100-{\displaystyle \frac{25{\theta }^2}{{\pi }^2}}}$.

Find the value of $\theta$ that gives the maximum volume, and find this maximum volume.

The maker wants a hat with a volume of 300 cm${}^3$. Find the two possible values of $\theta$.

Consider the equation $x^3-3x+c=0$, where $c\in \mathbb{R}$.

By sketching the graph of $y=x^3-3x+c$, or otherwise, write down the number of distinct real roots of the equation when $c=1$.

Write down the number of distinct real roots of the equation when $c=2$.

Write down the number of distinct real roots of the equation when $c=3$.

Factorise $x^3-3x+2$ completely.

Let $p(x)$ be a polynomial with a repeated root $x=a$, so that $p(x)=(x-a{)}^{2}g(x)$, where $g(x)$ is a polynomial.

Show that $p(a)=0$ and $p^{\mathrm{\prime }}(a)=0$.

Consider now the equation $x^3+px+q=0$, where $p,q\in \mathbb{R}$ and $p\mathrm{\ne }0$. Suppose that this equation has a repeated root $x=a$.

Use part (c) to show that $3a^2+p=0$.

Show that $a=-{\displaystyle \frac{3q}{2p}}$.

Hence show that $4p^3+27q^2=0$.

Verify that the equation in part (b) satisfies this condition.

Find the values of $k$ for which the equation $x^3-12x+k=0$ has a repeated root. For each of these values of $k$, find all the roots of the equation.

By using the substitution $x=y-2$, find the values of $c$ for which the equation $x^3+6x^2+3x+c=0$ has a repeated root.

Give your answers in the form $m\pm n\sqrt{3}$, where $m,n\in \mathbb{Z}$.

Suppose now that $p<0$. The graph of $y=x^3+px+q$ has stationary points where $x=\pm s$, where $s=\sqrt{-{\displaystyle \frac{p}{3}}}$.

Show that the product of the $y$-coordinates of the two stationary points is $q^2+{\displaystyle \frac{4p^3}{27}}$.

Hence state the condition on $p$ and $q$ for the equation $x^3+px+q=0$ to have three distinct real roots. Justify your answer.