1003230201
Level:
B
Given the function \( f(x)=x\cdot\,\mathrm{e}^x \), find the set of all \( x \), \( x\in\mathbb{R} \), for which \( f'(x)=0 \).
\( \{-1\} \)
\( \{-1;0\} \)
\( \{0;1\} \)
\( \emptyset \)
\( \{0\} \)
\( \{1\} \)
Derivative
1103163601
Level:
A
The graph of \( f \) is given in the figure. Choose which of the following graphs is the graph of \( f' \). (The function \( f' \) is the derivative of the function \( f \).)
1103143705
Level:
A
Choose the graph of a function \( f \) for which \( f'(-1) > 0 \), \( f'(0)=0 \), \( f'(1) < 0 \), \( f'(3) > 0 \) (\( f' \) is the derivative of the function \( f \)).
1103143704
Level:
A
Choose the graph of a function \( f \) for which \( f(-2)=0 \), \( f'(-1)=0 \), \( f'(0)=1 \), \( f'(2)=-3 \) (\( f' \) is the derivative of the function \( f \)).
1103143703
Level:
A
Choose the graph of a function \( f \) for which \( f'(-2)=0 \), \( f'(-1)=-2 \), \( f'(2)=4 \) (\( f' \) is the derivative of the function \( f \)).
1103143702
Level:
A
Choose the graph of a function \( f \) for which \( f(0)=0 \), \( f'(1)=-1 \), \( f'(2)=0 \), \( f'(3)=3 \) (\( f' \) is the derivative of the function \( f \)).
1103143701
Level:
A
Choose the graph of a function \( f \) for which \( f(0)=0 \), \( f'(0)=3 \), \( f'(1)=0 \), \( f'(2)=-1 \) (\( f' \) is the derivative of the function \( f \)).
1003112011
Level:
B
Differentiate the following function:
\[ f(x)=\frac{x-5\sqrt[3]{x^2}}{\sqrt[3]x} \]
\( f'(x)=\frac{2\sqrt[3]{x^2}-5\sqrt[3] x}{3x}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{2\sqrt[3]{x^2}-5\sqrt[3]x }{3}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{2\sqrt[3]x-5\sqrt[3]{x^2}}{3x}\text{, }x\in\mathbb{R}\setminus\{0\} \)
\( f'(x)=\frac{\sqrt[3]x}{3x}-5\text{, }x\in\mathbb{R}\setminus\{0\} \)
1003112010
Level:
B
Differentiate the following function:
\[ f(x)=\frac{\sqrt{x\sqrt x}}x \]
\( f'(x)=-\frac{\sqrt[4]{x^3}}{4x^2}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=-\frac{\sqrt[4]{x^3}}{x^2}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=-\frac{\sqrt[4]x}{4x}\text{, }x\in\mathbb{R}^+ \)
\( f'(x)=\frac{\sqrt[4]x}{4x}\text{, }x\in\mathbb{R}^+ \)
1003112009
Level:
A
Which of the statements A, B, C, D given bellow are incorrect?
\[
\begin{array}{l}
\text{A: }\ \left(\sqrt{x}-\cos x+\mathrm{ln}\, x\right)'=\frac1{2\sqrt x}+\sin x+\frac1x\text{, }x\in\mathbb{R}^+ \\
\text{B: }\ \left(3\sin x-3^x+x^3 \right)'=3\cos x-3^x+3x^2 \text{, }x\in\mathbb{R} \\
\text{C: }\ \left(\mathrm{cotg}\, x-\mathrm{log}_3 5\right)'=-\frac1{\sin^2 x}\text{, }x\in\mathbb{R}\setminus\bigcup\limits_{k\in\mathbb{Z}}\{k\pi\} \\
\text{D: }\ \left(3\sqrt[3]{x^2}-2\mathrm{e}^x \right)'=\frac2{\sqrt[3]x}-2\mathrm{e}^x\text{, }x\in\mathbb{R}\setminus\{0\}
\end{array}
\]
The only incorrect statements are:
B
B, C
A, B, C
C, D
B, C, D
C