2010013905
Level:
B
Suppose a function is concave down on the interval \([-1;2]\). Which one of the offered functions has that property?
\(g(x)=\sqrt{-2x+8}\)
\(f(x)=\frac{-x+2}{(x+3)^2}\)
\(h(x)=\frac{x^2-1}{x}\)
\(k(x)=\frac12x^3+3x^2-x+2\)
Analyzing function behavior
2110013904
Level:
A
The pictures show part of graphs of even functions. Choose the picture showing the part of the graph of the function \(f(x)=-\frac4{2+x^2}\).
2110013903
Level:
A
The pictures show parts of graphs of even functions. Choose the picture showing the part of the graph of the function \(f(x)=\frac4{1+x^2}\).
2110013902
Level:
A
The pictures show parts of graphs of functions that are decreasing on the interval \([1;5]\). Choose the picture showing the part of the graph of the function \[f(x)=\frac{x+7}{x+1}.\]
2110013901
Level:
A
The pictures show parts of graphs of functions that are increasing on the interval \([1;5]\). Choose the picture showing the part of the graph of the function \[f(x)=\frac{5x-1}{x+1}.\]
2010017803
Level:
A
Determine the values of \( a \) and \( b \) (\( a \), \( b \in\mathbb{R} \)) such that the function
\[ f(x)=ax^3-2bx+2 \]
has a local extremum of \( 6 \) at \( x=-1 \).
\( a=2 \), \( b=3 \)
\( a=-2 \), \( b=3 \)
\( a=-2 \), \( b=-3 \)
\( a=2 \), \( b=-3 \)
2010012505
Level:
A
Identify a true statement about the function
\(f(x) = -\frac{3}
{4}x^{4} +2x^{3}\).
The function \(f\) has
a local maximum at \(x = 2\).
The function \(f\) has
a local minimum at \(x = 0\).
The function \(f\)
has two local extrema. These extrema are at
\(x = 0\) and
\(x = 2\).
The function \(f\)
has neither local minimum nor local maximum.
2110012504
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(1) \text{ does not exist}; \\
f''(x) < 0 \text{ if } x < 1 ; \\
f''(x) < 0 \text{ if } x > 2; \\
f''(x) > 0 \text{ if } 1 < x < 2
\end{gather*}
($f'$ is the derivative of a function $f$, $f''$ is the second derivative of a function $f$).
2110012503
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(-2)=f'(0)=0; \\
f''(-2) < 0;\ f''(0) > 0
\end{gather*}
($f'$ is the derivative of the function $f$, $f''$ is the second derivative of the function $f$).
2010001703
Level:
A
Find the intervals where the following function is an increasing function.
\[
f(x) = \frac{4+x^{2}}
{-4x}
\]
\([ - 2;0)\) and
\((0;2] \)
\([ - 2;2] \)
\((-\infty ;-2] \) and
\([2;\infty) \)
\([ 2;\infty) \)