Problem 759
Using the Lagrange mean value theorem prove that if \(\displaystyle f\) is differentiable on \(\displaystyle \R\) and \(\displaystyle f'\) is bounded, then \(\displaystyle f\) is Lipschitz. Difficulty: 4. |
Problem 762
Using the Lagrange mean value theorem prove that if \(\displaystyle f'(a+0)\) exists, then \(\displaystyle f'_+(a)\) also exists and they are equal. Difficulty: 5. |
Problem 770
Let \(\displaystyle a_1<a_2<\ldots<a_n\) and \(\displaystyle b_1<b_2<\ldots<b_n\) be real numbers Show that \(\displaystyle \det\begin{pmatrix} e^{a_1b_1} & e^{a_1b_2} & \dots & e^{a_1b_n} \\ e^{a_2b_1} & e^{a_2b_2} & \dots & e^{a_2b_n} \\ \vdots & \vdots & \ddots & \vdots \\ e^{a_nb_1} & e^{a_nb_2} & \dots & e^{a_nb_n} \\ \end{pmatrix} >0. \) (KöMaL A. 463., October 2008) Difficulty: 9. Solution is available for this problem. |
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