Applications of derivatives

1003263404

Level: 
C
Find the global extrema of the following function on the interval \( [-1;3] \). \[ f(x)=x^2\cdot \mathrm{e}^{-x} \]
the global minimum at \( x=0 \), the global maximum at \( x=-1 \)
the global minimum at \( x=0 \), the global maximum at \( x=2 \)
the global minimum at \( x=3 \), the global maximum at \( x=-1 \)
the global minimum at \( x=-1 \), the global maximum at \( x=0 \)

1003263405

Level: 
C
Find a true statement about the function \( f(x)=\sin x+\frac12\cos⁡2x \) on the interval \( [0;\pi] \).
The function has global minima at the points \( x=0 \), \( x=\frac{\pi}2 \) and \( x=\pi \).
The only global minimum of \( f \) on this interval is at the point \( x=\frac{\pi}2 \).
The only global maximum of \( f \) on this interval is at the point \( x=\frac{\pi}6 \).
The function \( f \) has no global minimum on this interval.

1003266402

Level: 
C
The price of an Archery game experience program for groups up to $8$ participants is $12$ EUR/person. In case of a larger group (number of participants higher than $8$), each additional person reduces the price for all participants by $0.5$ $\mathrm{EUR}$/person. Find the number of participants that will bring the organizing company maximum income and calculate the total income.
There will be a maximum income of $128$ $\mathrm{EUR}$ for $16$ participants.
There will be a maximum income of $128$ $\mathrm{EUR}$ for $8$ participants.
There will be a maximum income of $192$ $\mathrm{EUR}$ for $16$ participants.
There will be a maximum income of $192$ $\mathrm{EUR}$ for $12$ participants.
None of the answers is correct.

1103263401

Level: 
C
The graph of \( f \) is given in the figure. Choose which of the following statements about the function \( f \) are true. \[ \begin{array}{l} \text{A: The global maximum of } f \text{ on the interval }[-4;4] \text{ is at } x=4. \\ \text{B: The only global minimum of } f \text{ on the interval } [-4;4] \text{ is at } x=2. \\ \text{C: On } (-2;3] \text{ there is the global minimum of } f \text{ at } x=2 \text{ and the global maximum of } f \text{ at } x=-2. \\ \text{D: The function } f \text{ has no global maximum on the interval } [-3;4). \\ \text{E: The function } f \text{ has no global minimum on the interval } [-4;2). \end{array} \] The only true statements are:
A, D
B, C
B, D, E
A, D, E
A, B, E
C, D

1103263402

Level: 
C
The graph of \( f \) is given in the figure. Choose which of the following statements about the function \( f \) are true. \[ \begin{array}{l} \text{A: The global minimum of } f \text{ on the interval } (-3;3) \text{ is at } x=0. \\ \text{B: The global maxima of } f \text{ on the interval } [-3;3] \text{ are at } x=-2 \text{ and } x=2. \\ \text{C: On } (-2;3] \text{ there is the global minimum of } f \text{ at } x=3 \text{ and the global maximum of } f \text{ at } x=2. \\ \text{D: The function } f \text{ has no global minimum on the interval } (-3;3). \\ \text{E: The function } f \text{ has no global maximum on the interval } (-3;3) . \end{array} \] The only true statements are:
B, C, D
C, D, E
A, B, C
A, B
C, D
A, E

1103266401

Level: 
C
A producer of sterilized canned vegetables needs to reduce the production costs of a $0.5$ liter cylindrical can. Find such radius $r$ and the height $h$ of the can (in centimeters) so that its surface (i.e. the amount of material needed) is minimal.
$r\doteq 4.3\,\mathrm{cm}$, $h\doteq 8.6\,\mathrm{cm}$
$r\doteq 3.4\,\mathrm{cm}$, $h\doteq 13.8\,\mathrm{cm}$
$r\doteq 5.4\,\mathrm{cm}$, $h\doteq 5.5\,\mathrm{cm}$
$r\doteq 3.4\,\mathrm{cm}$, $h\doteq 8.6\,\mathrm{cm}$

1103266403

Level: 
C
We want to create a rabbit cage in the shape of a rectangle with sides $a$ and $b$. The cage will be divided by parallel walls into four sections with the same area (see the picture). Find the dimensions $a$ and $b$ providing we have $50\,\mathrm{m}$ of fencing wire available and we want the total area to be as large as possible. (Fencing wire will also be used for the walls.)
$a=5\,\mathrm{m}$, $b=12.5\,\mathrm{m}$
$a=4\,\mathrm{m}$, $b=15\,\mathrm{m}$
$a=4.5\,\mathrm{m}$, $b=13.75\,\mathrm{m}$
$a=6.5\,\mathrm{m}$, $b=8.75\,\mathrm{m}$

1103266405

Level: 
C
Adam's House ($A$) is located at the distance of $0.9\,\mathrm{km}$ from the road. There is a bus stop ($B$) on this road at the distance of $1.5\,\mathrm{km}$ from the house (see the picture). Adam has overslept and needs to get to the bus stop as quickly as possible. At what distance $x$ from the nearest point $P$ should Adam reach the road knowing that he can move at the speed of $6\,\mathrm{km}/\mathrm{h}$ in rough terrain while being on the road he can move at the speed of $10\,\mathrm{km}/\mathrm{h}$?
$0.675\,\mathrm{km}$
$0.525\,\mathrm{km}$
$0.625\,\mathrm{km}$
$0.575\,\mathrm{km}$

1103266406

Level: 
C
The medieval builder has a $5$-ell-long iron belt. His task is to shape the belt into a frame of the Romanesque window (that is the union of a rectangle and a semicircle, see the picture). Find the optimal width $x$ of the window to get as much light coming through the window as possible (i. e. the area of the window should be as large as possible). Express the result rounded in inches ($1\,\mathrm{ell} = 45\,\mathrm{inches}$).
$63$
$140$
$32$
$112$
$83$
$20$