Applications of derivatives

9000079106

Level: 
A
Given function \(f(x)= x\mathrm{e}^{\frac{1} {x} }\), identify a true statement.
The local minimum of the function \(f\) is at the point \(x = 1\), the function does not have a local maximum.
The local maximum of the function \(f\) is at the point \(x = 0\), the local minimum at \(x = 1\).
The local maximum of the function \(f\) is at the point \(x = 1\), the function does not have a local minimum.
The function \(f\) has neither local minimum nor maximum.

1003263403

Level: 
B
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 \( x=0 \)

1003263404

Level: 
B
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: 
B
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.

1103163505

Level: 
B
Choose the graph of a function $f$ that satisfies \begin{gather*} f'(0) \text{ does not exist}; \\ f''(x) > 0 \text{ if } x < 0 ; \\ f''(x) > 0 \text{ if } x > 1; \\ f''(x) < 0 \text{ if } 0 < x < 1 \end{gather*} ($f'$ is the derivative of a function $f$, $f''$ is the second derivative of a function $f$).