SL 5.6—Stationary points, local max and min Questionbank

Practice SL 5.6—Stationary points, local max and min with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A function is defined by $f(x)=x^{5}-10 x^{3}+15 x^{2}+2 x-1$ , for $-3 \leq x \leq 3$ .

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

[5]

Determine the $x$ -coordinates of all stationary points and classify them using the second derivative test.

[6]

Find the $x$ -coordinates of the points of inflection by solving $f^{\prime \prime}(x)=0$ .

[5]

The region enclosed by $y=f(x)$ , the $x$ -axis, and the lines $x=-2$ and $x=2$ is rotated $360^{\circ}$ about the $y$ -axis. Set up and evaluate the integral to find the volume of the solid formed.

[5]

Question 2

HLPaper 1

The wind chill index $W$ is a measure of the temperature, in $^\circ\text{C}$ , felt when taking into account the effect of the wind.

When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind $v$ in kilometres per hour $(\text{km\,h}^{-1})$ is given by the equation

W = 19.34 - 7.405v^{0.16}

Find an expression for $\frac{dW}{dv}$ .

[2]

When Frieda arrives at the top of a hill, the speed of the wind is $10$ kilometres per hour and increasing at a rate of $5$ $\text{km\,h}^{-1}\,\text{min}^{-1}$ .

Find the rate of change of $W$ at this time.

[5]

Question 3

SLPaper 1

A software company is analyzing the performance of a new application, represented by the function $f(x) = x^3 - 6x^2 + 9x - 2$ , where $x$ represents the number of users (in thousands) and $f(x)$ represents the user satisfaction score.

Find the derivative $f^{\prime}(x)$ to assess how user satisfaction changes with the number of users.

[2]

Determine the $x$ -coordinates of the stationary points by setting $f^{\prime}(x) = 0$ .

[4]

Use the second derivative test to determine the nature of these stationary points.

[3]

Question 4

SLPaper 1

The height of a projectile, $h(t)$ meters, launched at time $t$ seconds, is modeled by $h(t) = -4.9t^2 + 19.6t + 2$ , for $t \geq 0$ .

Find $h'(t)$ .

[2]

Determine the time when the projectile reaches its maximum height.

[2]

Find the maximum height and verify it is a maximum using the second derivative.

[4]

Sketch the graph of $h(t)$ for $0 \leq t \leq 5$ , showing the maximum point and intercepts.

[4]

Question 5

SLPaper 1

A company's profit per year was found to be changing at a rate of $\frac{dP}{dt} = 3t^2 - 8t$ where $P$ is the company's profit in thousands of dollars and $t$ is the time since the company was founded, measured in years.

Determine whether the profit is increasing or decreasing when $t = 2$ .

[2]

One year after the company was founded, the profit was $4$ thousand dollars. Find an expression for $P(t)$ , when $t \ge 0$ .

[4]

Question 6

SLPaper 1

Consider the function $f(x) = x^2 - 4x + 3$ , where $x \in \mathbb{R}$ .

Find $f'(x)$ .

[2]

Determine the $x$ -coordinate of the stationary point.

[2]

Show that the stationary point is a local minimum.

[3]

Sketch the graph of $y = f(x)$ , indicating the stationary point and the intercepts with the axes.

[3]

Question 7

SLPaper 2

A cylindrical container with radius $r \text{ cm}$ and height $h \text{ cm}$ has a volume of $500 \text{ cm}^3$ . The surface area, $S$ , includes the top and bottom.

Express $h$ in terms of $r$ .

[2]

Show that the surface area is $S(r) = 2\pi r^{2} + \frac{1000}{r}$ .

[3]

Find $\frac{dS}{dr}$ and solve $\frac{dS}{dr} = 0$ .

[4]

Determine the minimum surface area and verify it is a minimum.

[3]

Question 8

SLPaper 2

A renewable energy company is evaluating the efficiency of a new solar panel design, represented by the function $f(x) = (x^3 + 2x^2)(x^2 - 1)$ .

Find the derivative $f'(x)$ using the product rule.

[4]

Solve for the values of $x$ where the slope of the curve is zero.

[3]

Determine whether the stationary points are maxima, minima, or points of inflection.

[2]

Question 9

SLPaper 2

A company's profit, $P(x)$ , in thousands of dollars, from selling $x$ units of a product is modeled by $P(x) = -x^{2} + 6x - 5$ , where $x \geq 0$ .

Find the derivative $P^{\prime}(x)$ .

[2]

Calculate the number of units $x$ that maximizes the profit.

[2]

Determine the maximum profit and interpret its meaning in the context of the problem.

[3]

Sketch the graph of $P(x)$ for $0 \leq x \leq 6$ , showing the stationary point and intercepts.

[3]

Question 10

SLPaper 1

A company is analyzing the performance of a new product, and the function $f$ representing its sales growth is given by its first derivative:

f^{\prime}(x) = 2x^2 - 5kx + 4,

where $k$ is a constant that reflects market conditions.

Calculate the second derivative $f^{\prime \prime}(x)$ .

[2]

The company finds that the graph of $f(x)$ has a point of inflection at $x = 1.5$ . Determine the value of $k$ .

[3]

Find $f^{\prime}(0)$ .

[2]

Determine the equation of the tangent line to the graph of $f$ at the point $(0, -2)$ .

[4]

Show that $f^{\prime}(1) = 0$ and hence determine the nature of the critical point at $x = 1$ .

[3]