Решение смотри в приложении
Точка, не примыкающая к отрезку.Встречается в нестрогих неравенствах
пример: [-5;2]v{6}
Задана функция
![f(x)=2x- \sqrt{x}](https://tex.z-dn.net/?f=f%28x%29%3D2x-+%5Csqrt%7Bx%7D+)
Область определения функции:
![D(f)=[0;+\infty)](https://tex.z-dn.net/?f=D%28f%29%3D%5B0%3B%2B%5Cinfty%29)
Найдем производную функции:
![f'(x)=(2x- \sqrt{x} )'=2- \frac{1}{2 \sqrt{x} }](https://tex.z-dn.net/?f=f%27%28x%29%3D%282x-+%5Csqrt%7Bx%7D+%29%27%3D2-+%5Cfrac%7B1%7D%7B2+%5Csqrt%7Bx%7D+%7D+)
Приравниваем функцию к нулю и находим критические точки
![2-\frac{1}{2 \sqrt{x} }=0|\cdot 2 \sqrt{x} \\ 4 \sqrt{x} -1=0\\ x= \frac{1}{16}](https://tex.z-dn.net/?f=2-%5Cfrac%7B1%7D%7B2+%5Csqrt%7Bx%7D+%7D%3D0%7C%5Ccdot+2+%5Csqrt%7Bx%7D+%5C%5C+4+%5Csqrt%7Bx%7D+-1%3D0%5C%5C+x%3D+%5Cfrac%7B1%7D%7B16%7D+)
Критические точки
![x= \frac{1}{16}](https://tex.z-dn.net/?f=x%3D+%5Cfrac%7B1%7D%7B16%7D)
[0]___-___(1/16)___+_____
![x=\frac{1}{16}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B1%7D%7B16%7D)
- Точка минимума, а точки максимума - нет.
1)
![(1+tg^{2}a)(1+ctg^{2}a)*tg^{2}a-(1-tg^{2}a)^{2}=(1+tg^{2}a)(1+ \frac{1}{tg^{2}a})*tg^{2}a-(1-2tg^{2}a+tg^{4}a)=(1+tg^{2}a)(tg^{2}a+1)-1+2tg^{2}a-tg^{4}a=(1+tg^{2}a)^{2}-1+2tg^{2}a-tg^{4}a=1+2tg^{2}a+tg^{4}a-1+2tg^{2}a-tg^{4}a=4tg^{2}a](https://tex.z-dn.net/?f=%281%2Btg%5E%7B2%7Da%29%281%2Bctg%5E%7B2%7Da%29%2Atg%5E%7B2%7Da-%281-tg%5E%7B2%7Da%29%5E%7B2%7D%3D%281%2Btg%5E%7B2%7Da%29%281%2B+%5Cfrac%7B1%7D%7Btg%5E%7B2%7Da%7D%29%2Atg%5E%7B2%7Da-%281-2tg%5E%7B2%7Da%2Btg%5E%7B4%7Da%29%3D%281%2Btg%5E%7B2%7Da%29%28tg%5E%7B2%7Da%2B1%29-1%2B2tg%5E%7B2%7Da-tg%5E%7B4%7Da%3D%281%2Btg%5E%7B2%7Da%29%5E%7B2%7D-1%2B2tg%5E%7B2%7Da-tg%5E%7B4%7Da%3D1%2B2tg%5E%7B2%7Da%2Btg%5E%7B4%7Da-1%2B2tg%5E%7B2%7Da-tg%5E%7B4%7Da%3D4tg%5E%7B2%7Da)
- что и требовалось доказать
2)
![(1+sin^{2}b)*ctg^{2}b- \frac{1}{sin^{2}b}=(1+sin^{2}b)*\frac{cos^{2}b}{sin^{2}b}- \frac{1}{sin^{2}b}=\frac{cos^{2}b}{sin^{2}b}+cos^{2}b- \frac{1}{sin^{2}b}=\frac{cos^{2}b-1}{sin^{2}b}+cos^{2}b=\frac{-(1-cos^{2}b)}{sin^{2}b}=\frac{-sin^{2}b}{sin^{2}b}=-1](https://tex.z-dn.net/?f=%281%2Bsin%5E%7B2%7Db%29%2Actg%5E%7B2%7Db-+%5Cfrac%7B1%7D%7Bsin%5E%7B2%7Db%7D%3D%281%2Bsin%5E%7B2%7Db%29%2A%5Cfrac%7Bcos%5E%7B2%7Db%7D%7Bsin%5E%7B2%7Db%7D-+%5Cfrac%7B1%7D%7Bsin%5E%7B2%7Db%7D%3D%5Cfrac%7Bcos%5E%7B2%7Db%7D%7Bsin%5E%7B2%7Db%7D%2Bcos%5E%7B2%7Db-+%5Cfrac%7B1%7D%7Bsin%5E%7B2%7Db%7D%3D%5Cfrac%7Bcos%5E%7B2%7Db-1%7D%7Bsin%5E%7B2%7Db%7D%2Bcos%5E%7B2%7Db%3D%5Cfrac%7B-%281-cos%5E%7B2%7Db%29%7D%7Bsin%5E%7B2%7Db%7D%3D%5Cfrac%7B-sin%5E%7B2%7Db%7D%7Bsin%5E%7B2%7Db%7D%3D-1)