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SL 5.6—Stationary points, local max and min - IB Questionbank | RevisionDojo

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

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

Find $h^{\prime}(t)$ .

[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 2

SLPaper 1

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

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

[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 3

SLPaper 2

A cylindrical container with radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$ has a volume of $500 \mathrm{~cm}^{3}$ . 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 $\frac{d S}{d r}$ and solve $\frac{d S}{d r}=0$ .

[4]

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

[3]

Question 4

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 5

SLPaper 2

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

[3]

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

[3]

Question 6

SLPaper 1

A function is defined by $f(x)=2 x^{3}-12 x^{2}+18 x+1$ , where $x \in \mathbb{R}$ .

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

[3]

Determine the $x$ -coordinates of the stationary points.

[3]

Classify each stationary point as a local maximum or minimum.

[3]

Sketch the graph of $y=f(x)$ for $-1 \leq x \leq 4$ , indicating the stationary points and the $y$ -intercept.

[3]

Question 7

SLPaper 1

A function is defined by $f(x)=\sin ^{2} x-\cos x+0.5$ , for $0 \leq x \leq 2 \pi$ .

Find $f^{\prime}(x)$ and simplify it using trigonometric identities.

[4]

Determine the $x$ -coordinates of all stationary points in $[0,2 \pi]$ .

[5]

Classify each stationary point as a maximum, minimum, or point of inflection, and find their $y$ -coordinates.

[6]

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

[5]

Sketch the graph of $y=f(x)$ for $0 \leq x \leq 2 \pi$ , indicating stationary points, inflection points, and intercepts.

[4]

The region bounded by $y=f(x)$ and the $x$ -axis where $f(x) \geq 0$ is rotated $360^{\circ}$ about the $x$ -axis. Find the volume of the solid formed.

[5]

Question 8

SLPaper 2

A rectangular field with a fixed perimeter of 80 meters has a length of $x$ meters and a width of $y$ meters. A fence runs parallel to the width, dividing the field into two equal areas.

Express $y$ in terms of $x$ .

[2]

Show that the total length of fencing, $F(x)$ , including the dividing fence, is given by $F(x)=3 x+\frac{80}{x}$ .

[3]

Find $\frac{d F}{d x}$ and solve $\frac{d F}{d x}=0$ to find the value of $x$ that minimizes the fencing.

[4]

Calculate the minimum fencing length and interpret the result.

[3]

Question 9

SLPaper 1

The height of a ball, $h(t)$ meters, after $t$ seconds is given by $h(t)=-5 t^{2}+20 t+1$ , for $t \geq 0$ .

Find the derivative $h^{\prime}(t)$ .

[2]

Determine the time $t$ when the ball reaches its maximum height.

[2]

Calculate the maximum height of the ball.

[2]

Sketch the graph of $h(t)$ for $0 \leq t \leq 4$ , indicating the maximum point and intercepts.

[4]

Question 10

SLPaper 2

A rectangular garden has a length of $x$ meters and a width of $y$ meters. The area of the garden is $24 \mathrm{~m}^{2}$ . A path of width 1 meter runs along the length of the garden, and its area is included in the total area.

Express $y$ in terms of $x$ .

[2]

The total fencing required, $F(x)$ , to enclose the garden is given by $F(x)=2 x+2 y$ . Show that $F(x)=2 x+\frac{48}{x}$ .

[3]

Find the derivative $\frac{d F}{d x}$ .

[2]

Find the value of $x$ that minimizes the fencing required and the minimum fencing length.

[3]

SL 5.6—Stationary points, local max and min Questionbank