Applications of derivatives

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.

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.
$r\doteq 4.3\,\mathrm{cm}$, $h\doteq 8.6\,\mathrm{cm}$
$r\doteq 3.4\,\mathrm{cm}$, $h\doteq 13.8\,\mathrm{cm}$
$r\doteq 5.4\,\mathrm{cm}$, $h\doteq 5.5\,\mathrm{cm}$
$r\doteq 3.4\,\mathrm{cm}$, $h\doteq 8.6\,\mathrm{cm}$

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.

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 \( x=0 \)

1003263403

Level: 
C
Find the global extrema of the following function on the interval \( [0;3] \). \[ f(x)=2x^3-3x^2-12x \]
the global minimum at \( x=2 \), the global maximum at \( x=0 \)
the global minimum at \( x=2 \), 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 \( x=0 \)

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

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