9000079102
Level:
A
Find the intervals where the following function is a decreasing function.
\[
f(x) = \frac{x^{2} + 1}
{x}
\]
\([ - 1;0)\) and
\((0;1] \)
\([ - 1;1] \)
\((-\infty ;-1] \) and
\([1;\infty) \)
\([1;\infty) \)
A
9000079103
Level:
A
Find the $x$ at which has the function $f$ a local maximum.
\[
f(x) = x^{3} - 3x^{2} - 9x + 2
\]
\(x=- 1\)
\(x=- 3\)
\(x=1\)
\(x=3\)
9000079104
Level:
A
Find the $x$ at which has the function $f$ a local minimum.
\[
f(x) = \frac{\ln x}
{x}
\]
does not exist
\(x = 0\)
\(x = 1\)
\(x =\mathrm{e}\)
9000079105
Level:
A
Find all the $x$ at which the function $f$ has local extrema.
\[
f(x)= \left (1 - x^{2}\right )^{3}
\]
\(x=0\)
\(x_1=0\),
\(x_2=1\)
\(x_1=- 1\),
\(x_2=1\)
\(x_1=- 1\),
\(x_2=0\),
\(x_3=1\)
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.
9000079203
Level:
A
Find all real \(x\)
for which the following expression equals zero.
\[
1 -\frac{2x + 1}
{x - 1}
\]
\(x = -2\)
\(x = -\frac{1}
{2}\)
\(x = 0\)
\(x = -1\)
9000079107
Level:
A
What is the function value of the function $f$ at its local minimum?
\[
f(x) = \frac{2}
{\sqrt{4x - x^{2}}}
\]
\(1\)
\(2\)
\(0\)
the local minimum does not exist
9000078501
Level:
A
Write the following set in an interval notation.
\[
\{x\in \mathbb{R};|x| > 2\}
\]
\((-\infty ;-2)\cup (2;\infty )\)
\([ 2;\infty ] \)
\((2;\infty )\)
\((-\infty ;-2] \cup [ 2;\infty )\)
9000078502
Level:
A
Write the following set in an interval notation.
\[
\{x\in \mathbb{R};|x|\leq 4\}
\]
\([ - 4;4] \)
\((-4;4)\)
\((-\infty ;-4] \)
\((-\infty ;-4)\)
9000073404
Level:
A
Find the sum of the following infinite series.
\[
\sqrt{2} - 2 + \sqrt{8} - 4 + \sqrt{32} - 8+\cdots
\]
The sum does not exist.
\(\frac{\sqrt{2}}
{1+\sqrt{2}}\)
\(\frac{\sqrt{2}}
{1-\sqrt{2}}\)
\(\sqrt{2} - 2\)