9000083706
Level:
A
Find all the values of \(x\in \mathbb{R}\)
for which the given expression equals zero.
\[
\frac{4x^{2} - 36}
{4x^{2} + 24x + 36}
\]
\(x = 3\)
\(x = 4\)
\(x = -3,\ x = 3\)
The expression never equals zero.
A
9000083707
Level:
A
Find all the values of \(x\in \mathbb{R}\)
for which the given expression equals zero.
\[
\frac{4x^{3} + 20x^{2} + 25x}
{x + 1}
\]
\(x = 0,\ x = -\frac{5}
{2}\)
\(x = 0\)
\(x = -\frac{5}
{2}\)
\(x = -1\)
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.
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.
\[
\{x\in \mathbb{R},|x|\leq 4\}
\]
\([ - 4,4] \)
\((-4,4)\)
\((-\infty ,-4] \)
\((-\infty ,-4)\)
9000078503
Level:
A
Write the following set in an interval notation.
\[
\{x\in \mathbb{R},|x - 3|\geq 5\}
\]
\((-\infty ,-2] \cup [ 8,\infty )\)
\((-\infty ,-8] \cup [ 2,\infty )\)
\([ 2,\infty )\)
\([ 8,\infty )\)
9000078504
Level:
A
Write the following set in an interval notation.
\[
\{x\in \mathbb{R},|x + 10| > 7\}
\]
\((-\infty ,-17)\cup (-3,\infty )\)
\((-\infty ,3)\cup (17,\infty )\)
\((-3,\infty )\)
\((17,\infty )\)
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 \(\left (\frac{5}
{2},\infty \right )\).