Problem 779
Which of the right circular cones inscribed into the unit sphere has the greatest volume? Difficulty: 2. |
Problem 780
Calculate the extremal values of the following functions on the given interval! (1) \(\displaystyle x^2 -x^4 ; \ [-2,2];\) (2) \(\displaystyle x-\arctg x; \ [-1,1];\) (3) \(\displaystyle x+e^{-x}; \ [-1,1];\) (4) \(\displaystyle x+x^{-2}; \ [1/10, 10];\) (5) \(\displaystyle \arctg(1/x); \ [1/10, 10];\) (6) \(\displaystyle \cos x^2 ;\ [0,\pi ];\) (7) \(\displaystyle \sin (\sin x); \ [-\pi /2,\pi /2];\) (8) \(\displaystyle x\cdot e^{-x} ;\ [-2,2];\) (9) \(\displaystyle x^n \cdot e^{-x}; \ [-2n,2n];\) (10) \(\displaystyle x-\log x; \ [1/2, 2];\) (11) \(\displaystyle 1/(1+\sin ^2 x),\ (0,\pi ) ;\) (12) \(\displaystyle \sqrt{1-e^{-x^2}}; \ [-2,2];\) (13) \(\displaystyle x\cdot \sin (\log x);\ [1,100];\) (14) \(\displaystyle x^x;\ (0,\infty );\) (15) \(\displaystyle \sqrt[x]{x};\ (0,\infty );\) (16) \(\displaystyle (\log x)/x;\ (0,\infty );\) (17) \(\displaystyle x\cdot \log x; \ (0,\infty );\) (18) \(\displaystyle x^x \cdot (1-x)^{1-x} ; \ (0,1).\) Difficulty: 2. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |