2010002107
Level:
A
Given the function \(f(x)=-5(x-1)^3+3(x-1)^5-2\) (see the picture), at how many points has this function null derivative?
\(3\)
\(2\)
\(1\)
\(4\)
Derivative
2010002106
Level:
A
How many of the following functions have the positive derivative at the point \(x = 3\)?
\(2\)
\(1\)
\(3\)
\(4\)
2010002105
Level:
A
How many of the following functions have the negative derivative at the point \(x = 1\)?
\(2\)
\(1\)
\(3\)
\(4\)
2110002104
Level:
A
In one of the following pictures there is a graph of a function, which is not differentiable at the point \(x = 3\). Choose this picture.
2110002103
Level:
A
In one of the following pictures there is a graph of a function, which is not differentiable at the point \(x = 1\). Choose this picture.
2110002102
Level:
A
In one of the following pictures there is a graph of a function, which has null derivative at the point \(x = 3\). Choose this picture.
2110002101
Level:
A
In one of the following pictures there is a graph of a function, which has null derivative at the point \(x = 1\). Choose this picture.
2010002009
Level:
B
Differentiate the following function.
\[
f(x) =\ln \left (\frac{2x}
{2 - x}\right )
\]
\(f^{\prime}(x) = \frac{2}
{(2-x)x} ;\ x\in \left (0;2\right )\)
\(f^{\prime}(x) = \frac{2}
{(2-x)x} ;\ x\in \mathbb{R}\setminus \left \{0;2\right \}\)
\(f^{\prime}(x) = \frac{2-x}
{2x};\ x\in \left (0;2\right )\)
\(f^{\prime}(x) = \frac{2-x}
{2x};\ x\in \mathbb{R}\setminus \left \{0;2\right \}\)
2010002008
Level:
B
Differentiate the following function.
\[
f(x) =\ln \left(3x^{2} - 5x \right)
\]
\(f^{\prime}(x) = \frac{6x-5}
{3x^{2}-5x};\ x\in \left (-\infty ;0\right )\cup \left (\frac{5}{3};\infty \right )\)
\(f^{\prime}(x) = \frac{6x-5}
{3x^{2}-5x};\ x\in \mathbb{R}\setminus \left \{0;\frac{5}
{3}\right \}\)
\(f^{\prime}(x) = \frac{1}
{3x^{2}-5x};\ x\in \left (-\infty ;0\right )\cup \left (\frac{5}{3};\infty \right )\)
\(f^{\prime}(x) = \frac{1}
{3x^{2}-5x};\ x\in \mathbb{R}\setminus \left \{0;\frac{5}
{3}\right \}\)
2010002007
Level:
B
Differentiate the following function.
\[
f(x) = \frac{\sqrt{x} +2}
{2-\sqrt{x} }
\]
\(f'(x) = \frac{2}
{\sqrt{x}}\frac{1}{(2-\sqrt{x})^{2}}, \ x \in (0 ;4)\cup
(4;\infty) \)
\(f'(x) = \frac{\sqrt{x}}
{2(2-\sqrt{x})^2}, \ x \in (0 ;4) \cup
(4;\infty) \)
\(f'(x) = \frac{1}
{(2-\sqrt{x})^2}, \ x \in (0 ;4) \cup
(4;\infty) \)
\(f'(x) = \frac{1}
{x(2-\sqrt{x})^2}, \ x \in (0 ;4) \cup
(4;\infty) \)