Analyzing function behavior

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.

9000145410

Level: 
A
Identify a true statement about the function \(f(x) = \frac{1} {4}x^{4} - x^{3}\).
The local minimum of \(f\) is at \(x = 3\).
The function \(f\) has neither local minimum nor local maximum.
The function \(f\) has a local minimum at \(x = 0\).
The function \(f\) has two local extrema. These extrema are at \(x = 3\) and \(x = 0\).

1003261904

Level: 
B
Given the function \[ f(x)=\sin ⁡x-3\cos⁡ x\text{ ,} \] determine the set of all \( x \), \( x\in\mathbb{R} \), such that \( f''(x)+f(x)=0 \).
\( \mathbb{R} \)
\( \emptyset \)
\( \{k\pi;\ k\in\mathbb{Z}\} \)
\( \left\{(2k+1)\frac{\pi}2;\ k\in\mathbb{Z} \right\} \)