85742 Minimum distance between the curves \(y^{2}=4 x\) and \(x^{2}+y^{2}-12 x+31=0\) is

85743 The minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma, \quad\) where \(\alpha, \beta, \gamma\) are real number satisfying \(\alpha+\beta+\boldsymbol{\gamma}=\boldsymbol{\pi}\) is

85744 The maximum value of
\(\left(\cos \alpha_{1}\right) \cdot\left(\cos \alpha_{2}\right) \ldots\left(\cos \alpha_{n}\right)\) Under the
restrictions \(0 \leq \alpha_{1}, \alpha_{2}, \ldots . . \alpha_{n} \leq \frac{\pi}{2}\) and
\(\left(\cot \alpha_{1}\right) \cdot\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1\) is

85745 \(\mathbf{f}(\mathbf{x})=\left\{\begin{array}{cc}\left|\mathbf{x}^3+\mathbf{x}^2+\mathbf{3 x}+\sin x\right| \cdot\left(3+\sin \frac{1}{x}\right) & , x \neq 0 \\ 0, & x=0\end{array}\right.\)
The number of points, where \(f(x)\) attains its minimum value, \(i\)

85742 Minimum distance between the curves \(y^{2}=4 x\) and \(x^{2}+y^{2}-12 x+31=0\) is

85743 The minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma, \quad\) where \(\alpha, \beta, \gamma\) are real number satisfying \(\alpha+\beta+\boldsymbol{\gamma}=\boldsymbol{\pi}\) is

85744 The maximum value of
\(\left(\cos \alpha_{1}\right) \cdot\left(\cos \alpha_{2}\right) \ldots\left(\cos \alpha_{n}\right)\) Under the
restrictions \(0 \leq \alpha_{1}, \alpha_{2}, \ldots . . \alpha_{n} \leq \frac{\pi}{2}\) and
\(\left(\cot \alpha_{1}\right) \cdot\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1\) is

85745 \(\mathbf{f}(\mathbf{x})=\left\{\begin{array}{cc}\left|\mathbf{x}^3+\mathbf{x}^2+\mathbf{3 x}+\sin x\right| \cdot\left(3+\sin \frac{1}{x}\right) & , x \neq 0 \\ 0, & x=0\end{array}\right.\)
The number of points, where \(f(x)\) attains its minimum value, \(i\)

85742 Minimum distance between the curves \(y^{2}=4 x\) and \(x^{2}+y^{2}-12 x+31=0\) is

85743 The minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma, \quad\) where \(\alpha, \beta, \gamma\) are real number satisfying \(\alpha+\beta+\boldsymbol{\gamma}=\boldsymbol{\pi}\) is

85744 The maximum value of
\(\left(\cos \alpha_{1}\right) \cdot\left(\cos \alpha_{2}\right) \ldots\left(\cos \alpha_{n}\right)\) Under the
restrictions \(0 \leq \alpha_{1}, \alpha_{2}, \ldots . . \alpha_{n} \leq \frac{\pi}{2}\) and
\(\left(\cot \alpha_{1}\right) \cdot\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1\) is

85745 \(\mathbf{f}(\mathbf{x})=\left\{\begin{array}{cc}\left|\mathbf{x}^3+\mathbf{x}^2+\mathbf{3 x}+\sin x\right| \cdot\left(3+\sin \frac{1}{x}\right) & , x \neq 0 \\ 0, & x=0\end{array}\right.\)
The number of points, where \(f(x)\) attains its minimum value, \(i\)

85742 Minimum distance between the curves \(y^{2}=4 x\) and \(x^{2}+y^{2}-12 x+31=0\) is

85743 The minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma, \quad\) where \(\alpha, \beta, \gamma\) are real number satisfying \(\alpha+\beta+\boldsymbol{\gamma}=\boldsymbol{\pi}\) is

85744 The maximum value of
\(\left(\cos \alpha_{1}\right) \cdot\left(\cos \alpha_{2}\right) \ldots\left(\cos \alpha_{n}\right)\) Under the
restrictions \(0 \leq \alpha_{1}, \alpha_{2}, \ldots . . \alpha_{n} \leq \frac{\pi}{2}\) and
\(\left(\cot \alpha_{1}\right) \cdot\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1\) is

85745 \(\mathbf{f}(\mathbf{x})=\left\{\begin{array}{cc}\left|\mathbf{x}^3+\mathbf{x}^2+\mathbf{3 x}+\sin x\right| \cdot\left(3+\sin \frac{1}{x}\right) & , x \neq 0 \\ 0, & x=0\end{array}\right.\)
The number of points, where \(f(x)\) attains its minimum value, \(i\)