1003261901
Level:
A
Find the second derivative of the function
\[ f(x)=\frac{x^2}{1-x} \]
at the point \( x_0=2 \).
\( -2 \)
\( 2 \)
\( -\frac14 \)
\( \frac14 \)
\( -4 \)
\( 4 \)
Applications of derivatives
1003261902
Level:
A
Find the second derivative of the function
\[ f(x)=\sin^2 x \]
at the point \( x_0=-\frac{\pi}6 \).
\( 1 \)
\( \frac12 \)
\( -\frac12 \)
\( -1 \)
\( \sqrt3 \)
\( -\frac{\sqrt3}2 \)
1003261903
Level:
A
Given the function
\[ f(x)=x^3-3x^2+2\text{ ,} \]
determine the set of all \( x \), \( x\in\mathbb{R} \), such that \( f''(x)-f'(x)=3 \).
\( \{1;3\} \)
\( \{-1;-3\} \)
\( \{-\sqrt3;\sqrt3\} \)
\( \{\sqrt3\} \)
\( \emptyset \)
1003261904
Level:
A
Given the function
\[ f(x)=\sin x-3\cos x\text{ ,} \]
determine the set of all \( x \), \( x\in\mathbb{R} \), such that \( f''(x)+f(x)=0 \).
\( \mathbb{R} \)
\( \emptyset \)
\( \{k\pi;\ k\in\mathbb{Z}\} \)
\( \left\{(2k+1)\frac{\pi}2;\ k\in\mathbb{Z} \right\} \)
1003261905
Level:
A
Find the local extrema of the function
\[ f(x)=x-\ln(1+x)\text{ .} \]
the local minimum at \( x=0 \)
the local minimum at \( x=0 \), the local maximum at \( x=-1 \)
the local maximum at \( x=0 \)
the local maximum at \( x=0 \), the local minimum at \( x=-1 \)
do not exist
1103163602
Level:
A
The graph of \( f' \) is given in the figure. Find the interval where \( f \) is an increasing function. (The function \( f' \) is the derivative of the function \( f \).)
\( (-1;1) \)
\( (-3;-1) \)
\( (2;4) \)
\( (0;2) \)
1103163603
Level:
A
The graph of \( f' \) is given in the figure. Find the interval where \( f \) is an increasing function. (The function \( f' \) is the derivative of the function \( f \).)
\( (1;2) \)
\( (-1;1) \)
\( (1;3) \)
\( (-1;0) \)
1103163604
Level:
A
The graph of \( f' \) is given in the figure. Find the interval where \( f \) is a decreasing function. (The function \( f' \) is the derivative of the function \( f \).)
\( (-3;-2) \)
\( (-1;1) \)
\( (0;2) \)
\( (-1;2) \)
1103163605
Level:
A
The graph of \( f' \) is given in the figure. Find the interval where \( f \) is a decreasing function. (The function \( f' \) is the derivative of the function \( f \).)
\( (2;4) \)
\( (-1;1) \)
\( (1;3) \)
\( (-4;-2) \)
1103163606
Level:
A
The graph of \( f' \) is given in the figure. Find the local extrema of \( f \). (The function \( f' \) is the derivative of the function \( f \).)
local minimum at \( x=0 \), local maxima at \( x_1=-2 \) and \( x_2=3 \)
local minimum at \( x=-1 \), local maximum at \( x=2 \)
local minima at \( x_1=-2 \) and \( x_2=3 \), local maximum at \( x=0 \)
local minima at \( x_1=-2 \) and \( x_2=0 \), local maximum at \( x=3 \)
local minimum at \( x=-2 \), local maxima at \( x_1=0 \) and \( x_2=2 \)