Second derivative, points of inflection, testing max / min points, optimisation: notes and practice questions

Consider the function $f$, which has the derivative $f^{'}(x) = - 4x^{2} + 28x + c,$ where $c\mathbb{\in Z}$.

$a$ Knowing that the y coordinate of the vertex of $f^{'}$ is equal to 9, find the value of $c$.

$b$ For what values of $x$ is the graph of $f$ concaved upwards. Justify your answer.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

A function, $g$, has its derivative given by $g'(x) = -2x^2 + 8x + k$, where $k \in \mathbb{R}$. The following diagram shows part of the graph of $g'$.

The graph of $g'$ has an axis of symmetry $x = q$.

(a) Find the value of $q$.

(b) The vertex of the graph of $g'$ has a y-coordinate of 10. Find the value of $k$.

(c) Find the equation of the tangent to the graph of $g'$ at $x = 0$.

The graph of $g$ has a point of inflexion at $x = c$.

(d) (i) Find the value of $c$.

(ii) Find the values of $x$ for which the graph of $g$ is concave-up. Justify your answer.

A ladder must be placed against a tall vertical building, clearing a monument that is 8 m high and stands on horizontal ground 1 m away from the building's base. The ladder touches the ground, the top corner of the monument, and the wall of the building.

Let $L$ be the length of the ladder in metres.

Let $\theta$ be the angle that the ladder makes with the ground, where $0 < \theta < \frac{\pi}{2}$.

(a) Show that $L = \sec \theta + 8\csc \theta$.

(b) (i) Find $\frac{dL}{d\theta}$.

(b) (ii) When $\frac{dL}{d\theta} = 0$, show that $\tan\theta = 2$.

(c) (i) Find $\frac{d^2L}{d\theta^2}$.

(c) (ii) When $\tan\theta = 2$, find the value of $\frac{d^2L}{d\theta^2}$.

(d) (i) Hence, justify that $L$ is a minimum when $\tan\theta = 2$.

(d) (ii) Determine this minimum value of $L$.

(e) A construction company only has ladders with a maximum length of 11 m. Determine whether it is possible to position a ladder against the building over the monument, giving a reason for your answer.

A function $f$ is defined by $f(x) = \frac{\ln(x)}{x}$, for $x > 0$.

The following diagram shows part of the graph of $f$.

(a) Find the coordinates of the x-intercept of the graph of $f$.

(b) Find $f'(x)$.

The graph of $f$ has a local maximum at point M.

(c) Hence, find the exact coordinates of M.

(d) (i) Show that $f''(x) = \frac{2\ln(x) - 3}{x^3}$.

The graph of $f$ has a point of inflection at point P.

(d) (ii) Hence, find the exact coordinates of P.

A rectangle is inscribed in an ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The sides of the rectangle are parallel to the coordinate axes. The vertices of the rectangle are located at $(\pm x, \pm y)$, where $x > 0$ and $y > 0$.

(a) Show that the area of the rectangle, $A$, can be expressed as $A = \frac{12x}{5}\sqrt{25-x^2}$.

(b) Show that $\frac{dA}{dx} = \frac{12(25-2x^2)}{5\sqrt{25-x^2}}$.

(c) Hence, find the exact dimensions of the rectangle with the maximum possible area.

A drone takes off from a platform. Its height, $h$ metres, above the platform after $t$ seconds is given by $h(t) = 6t - t^2$, for $0 \le t \le 8$. This is shown in the following diagram.

The drone lands back on the platform when $t=p$.

Find the value of $p$.

The drone reaches its maximum height when $t=q$.

Find the value of $q$.

Find the drone's maximum height above the platform.

Find the drone's vertical distance from the platform when $t=8$.

The total vertical distance travelled by the drone in the first 8 seconds is given by $d$.

Find the value of $d$.

A second drone, Drone B, takes off from the same platform. Its velocity is given by $v_B(t) = 8 - 2t$, for $t \ge 0$.

When $t = k$, the total vertical distance travelled by Drone B is equal to $d$.

Find the value of $k$.

A pharmaceutical company is testing a new drug. The concentration of the drug in a patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modeled by the function $C(t) = 10t \text{e}^{-0.5t}$, for $0 \le t \le 10$.

Sketch the graph of $C(t)$ on the grid below.

(b) Find the time, in hours, at which the concentration of the drug in the bloodstream is at its maximum.

A hollow pipe is manufactured by removing a smaller cylinder of radius $r$ from the centre of a larger cylinder of radius $3r$. Both cylinders have the same height, $h$. This is shown in the following diagram.

All lengths are measured in centimetres.

The total surface area of the hollow pipe, in cm$^2$, is given by $S$.

(a) Show that $S = 16\pi r^2 + 8\pi rh$.

The total surface area of the hollow pipe is $128\pi \text{ cm}^2$.

(b) Show that the volume of the pipe, $V$, is given by $V = 128\pi r - 16\pi r^3$.

(c) Find an expression for $\frac{dV}{dr}$.

(d) The hollow pipe has its maximum volume when $r = k\sqrt{6}$, where $k \in \mathbb{Q}^+$. Find the value of $k$.

(e) Hence, find this maximum volume, giving your answer in the form $m\pi\sqrt{6}$, where $m \in \mathbb{Q}^+$.

Consider the function $f(x) = e^x - 3x - 6$.

On the following axes, sketch the graph of $f$ for $-3 \le x \le 3$.

The function $g$ is defined by $g(x) = e^{2x} - 6x - 10$.

The graph of $g$ is obtained from the graph of $f$ (from part a) by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

Find the value of $k$ and the value of $c$.

A particle moves in a straight line. Its velocity, $v \,\text{ms}^{-1}$, at time $t$ seconds is given by $v(t) = -t^3 + 6t^2 - 9t$, for $0 \le t \le 5$. The particle is at the origin at $t=0$.

The graph of $v$ is shown in the following diagram.

(a) Find the displacement of the particle from the origin at $t=3$.

(b) Find an expression for the acceleration of the particle.

(c) The particle is momentarily at rest at $t=0$ and again at $t=k$. Find the greatest speed of the particle in the interval $0 \le t \le k$.

(d) Find the greatest speed of the particle for $0 \le t \le 5$.

(e) Write down an expression that represents the distance travelled by the particle while its speed is increasing. Do not evaluate the expression.

An engineer is designing an open-top rectangular container with a square base. The container must have a volume of $4\text{ m}^3$.

Let the side length of the square base be $x$ metres and the height of the container be $h$ metres.

Show that the total surface area, $S\text{ m}^2$, of the material used for the container is given by $S = x^2 + \frac{16}{x}$.

Find an expression for $\frac{dS}{dx}$.

Hence, find the exact value of $x$ for which the surface area is a local minimum or maximum.

Find an expression for $\frac{d^2S}{dx^2}$.

Use the second derivative of $S$ to justify that $S$ is a minimum when $x = 2$.

Find the minimum surface area of the container.

A civil engineer is analyzing the structural integrity of a new bridge design. The deflection of a certain point on the bridge, $D(x)$, in millimeters, is modeled by the function $D(x) = \frac{2x+1}{x^2-4}$, where $x$ represents the horizontal distance in meters from a central support. The model is valid for $x \in \mathbb{R}$, $x\neq p$, $x\neq q$.

Find the value of $p$ and the value of $q$.

Find an expression for $D'(x)$.

The graph of $y = D(x)$ has exactly one point of inflexion.

Find the x-coordinate of the point of inflexion.

Sketch the graph of $y = D(x)$ for $-4 \leq x \leq 4$, showing the values of any axes intercepts, the coordinates of any local maxima and local minima (if they exist), and giving the equations of any asymptotes.

Consider a related model for stress distribution, $S(x) = \frac{x^2-4}{2x+1}$ for $x \in \mathbb{R}$, $x \neq -\frac{1}{2}$.

Find the equations of all the asymptotes on the graph of $y = S(x)$.

The engineer needs to identify the regions where the bridge deflection $D(x)$ is less than $1$ mm. Solve $D(x) < 1$ for $x \in \mathbb{R}$.

A packaging company is designing a new cylindrical can. The can must have a fixed volume of $54\pi$ cm$^3$. The company wants to minimize the amount of material used, which corresponds to minimizing the total surface area of the can.

Let the radius of the can be $r$ cm and its height be $h$ cm.

Show that the total surface area, $S$ cm$^2$, of the can is given by $S = 2\pi r^2 + \frac{108\pi}{r}$.

The total surface area of the can has a local minimum value when $r = a$.

(i) Find an expression for $\frac{dS}{dr}$.

(ii) Hence, find the exact value of $a$.

(i) Find an expression for $\frac{d^2S}{dr^2}$.

(ii) Use the second derivative of $S$ to justify that $S$ is a minimum when $r = a$.

(iii) Find the minimum surface area of the can.

Two drones, Drone X and Drone Y, have position vectors with respect to an origin O given respectively by

$\boldsymbol{r}_X = \begin{pmatrix} 10 \\ -2 \\ 2 \end{pmatrix} + t \begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}$

$\boldsymbol{r}_Y = \begin{pmatrix} -1 \\ -2 \\ 9 \end{pmatrix} + t \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}$

where $t$ represents the time in minutes and $0 \le t \le 3$.

Entries in each column vector give the displacement east of O, the displacement north of O and the distance above sea level, all measured in kilometres.

(a) Find the three-figure bearing on which Drone Y is travelling.

(b) Show that Drone X travels at a greater speed than Drone Y.

(c) Find the acute angle between the two drones' lines of flight. Give your answer in degrees.

The two drones' lines of flight cross at point P.

(d) (i) Find the coordinates of P.

(ii) Determine the length of time between the first drone arriving at P and the second drone arriving at P.

(e) Let $D(t)$ represent the distance between Drone X and Drone Y for $0 \le t \le 3$.

Find the minimum value of $D(t)$.

A landscape architect is designing a triangular shade sail for a patio. The vertices of the sail are defined by points A, B, and C in a 3D coordinate system, where the z-axis represents height.

The coordinates of the vertices are A$(0, k, 1)$, B$(2, 1, 0)$ and C$(k, 0, 3)$, where $k$ is a positive constant representing a design parameter.

(a) Show that the vector product $\vec{AB} \times \vec{AC}$ is given by $\begin{pmatrix} 2-3k \\ -k-4 \\ k^2-3k \end{pmatrix}$.

(b) The architect wants to minimize the tension in the sail, which is proportional to the magnitude of the vector product of two adjacent sides. Find the smallest possible value of $|\vec{AB} \times \vec{AC}|$.

(c) Calculate the smallest possible area of the shade sail.

The growth of a bacterial colony, $B$, in a petri dish can be modelled by the logistic differential equation

$\frac{\text{d}B}{\text{d}t} = k B \left(1 - \frac{B}{N}\right)$

where $t$ is the time measured in hours and $k, N$ are positive constants.

The constant $N$ represents the maximum number of bacteria the petri dish can sustain indefinitely due to limited nutrients.

In the context of this bacterial growth model, interpret the meaning of $\frac{\text{d}B}{\text{d}t}$.

Show that $\frac{\text{d}^2B}{\text{d}t^2} = k^2B\left(1-\frac{B}{N}\right)\left(1-\frac{2B}{N}\right)$.

Hence show that the bacterial colony will grow at its maximum rate when $B = \frac{N}{2}$. Justify your answer.

Hence determine the maximum value of $\frac{\text{d}B}{\text{d}t}$ in terms of $k$ and $N$.

Let $B_0$ be the initial number of bacteria.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt = \ln\left(\frac{B(N-B_0)}{B_0(N-B)}\right)$.

After 5 hours, the number of bacteria is $2B_0$. It is known that $N = 3B_0$.

Find the value of $k$ for this bacterial growth model.

A toy rocket is launched vertically upwards from a platform. Its height, $h$ metres above the ground, $t$ seconds after launch, is modelled by the function $h(t) = 10 + 25t - 4.9t^2$.

Find the initial height of the platform from which the rocket is launched.

Find the maximum height reached by the rocket.

Calculate the time it takes for the rocket to hit the ground.

Write down the domain of the function $h$ in the context of this real-life scenario.

Determine the length of time for which the height of the rocket is greater than $30 \text{ m}$.