9000083703
Level:
A
Find all the values of \(x\in \mathbb{R}\)
for which the given expression equals zero.
\[
\frac{x^{3} - x}
{x - 1}
\]
\(x = -1,\ x = 0\)
\(x = 0\)
\(x = 1\)
\(x = -1,\ x = 0,\ x = 1\)
A
9000083704
Level:
A
Find all the values of \(x\in \mathbb{R}\)
for which the given expression equals zero.
\[
\frac{x^{2} - 4x + 4}
{x(x - 2)}
\]
The expression never equals zero.
\(x = 0\)
\(x = 2\)
\(x = -2,\ x = 0\)
9000079101
Level:
A
Find the intervals of monotonicity for the following function.
\[
f(x)= \frac{3x + 1}
{2x - 5}
\]
Decreasing on \(\left (-\infty ; \frac{5}
{2}\right )\)
and \(\left (\frac{5}
{2};\infty \right )\).
Decreasing on \(\left (-\infty ; \frac{5}
{2}\right )\cup \left (\frac{5}
{2};\infty \right )\).
Decreasing on \(\left (-\infty ; \frac{5}
{2}\right )\),
increasing on \(\left (\frac{5}
{2};\infty \right )\).
Increasing on \(\left (-\infty ; \frac{5}
{2}\right )\),
decreasing on \(\left (\frac{5}
{2};\infty \right )\).
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
\([1;\infty) \)
\([1;\infty) \)
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