2010010909
Level:
B
Find the maximum of the function \(g(x)=\frac{x^4}2-4x^2+2\) when \( x \in [ -2;3]\).
\(x=3\)
\(x=-2\)
\(x=0\)
The function \(g\) does not have the maximum on the interval \([ -2;3]\).
Applications of derivatives
2010010910
Level:
B
Find the minimum of the function \(g(x)=-{x^4}+2x^2+1\) when \( x \in [ -1;2]\).
\(x=2\)
\(x=-1\)
\(x=0\)
The function \(g\) does not have the minimum on the interval \([ -1;2]\).
2010012501
Level:
B
Find the global extrema of the following function on the interval \( [ 0;2 ] \).
\[ f(x)=x^3+3x^2-9x \]
the global minimum at \( x=1 \), the global maximum at \( x=2 \)
the global minimum at \( x=1 \), the global maximum at \( x=-3 \)
the global minimum at \( x=2 \), the global maximum at \( x=1 \)
the global minimum at \( x=0 \), the global maximum at \( x=2 \)
2010012502
Level:
B
Identify a true statement about the function
\(f(x) = x^{3} +6x^{2} + 12x -1\).
There is neither local minimum nor maximum of \(f\).
The function \(f\) has a local maximum at the point \(x = -2\).
The function \(f\) has a local minimum at the point \(x = -2\).
The global minimum of \(f\) on \(\mathbb{R}\) is at \(x = -2\).
2010012505
Level:
B
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.
2010013905
Level:
B
Suppose a function is concave down on the interval \([-1;2]\). Which one of the offered functions has that property?
\(g(x)=\sqrt{-2x+8}\)
\(f(x)=\frac{-x+2}{(x+3)^2}\)
\(h(x)=\frac{x^2-1}{x}\)
\(k(x)=\frac12x^3+3x^2-x+2\)
2010013906
Level:
B
Suppose a function is convex up on the interval \([-2;1]\). Which one of the offered functions has that property?
\(h(x)=-\sqrt{x+5}\)
\(f(x)=\frac{x-2}{(x+5)^2}\)
\(g(x)=\frac{x^2-2}{2x}\)
\(k(x)=-\frac15x^3-2x^2+x+1\)
2010013907
Level:
B
How many of the given functions have exactly one inflection point?
\[f(x)=(x+2)^5+(2+x)^3-2,\quad g(x)=\frac1{6(x+4)^4},\quad h(x)=\frac{x^3+2x^2+x+2}{x}\]
2
1
3
None of these functions has exactly one inflection point.
2010013908
Level:
B
How many of the given functions have exactly one inflection point?
\[f(x)=\frac1{-2(x+4)^4}+6,\quad g(x)=-(x-3)^5-(-3+x)^3+1,\quad h(x)=\frac{x^3-3x^2+6x+9}{-3x}\]
2
1
3
None of these functions has exactly one inflection point.
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 $f$).
