4а^2 + 12а + 9^2
Формула: (а+б)^2 = а^2 + 2аб + б^2
№1
(1+5x)(5x-1)-10(x-2)=19
25x^2-1-10x+20=19
25x^2-10x=0
5x(5x-2)=0
5x=0:5x-2
5x=0
x=0
![y=x+ \frac{4}{x}](https://tex.z-dn.net/?f=y%3Dx%2B+%5Cfrac%7B4%7D%7Bx%7D+)
Найти наибольшее и наименьшее значение функциии
на отрезке
![[1,4]](https://tex.z-dn.net/?f=%5B1%2C4%5D)
Вычислим значение функции в критических точках:
![f'=(x+ \frac{4}{x})'=x'+ \frac{4'\cdot x-4\cdot x'}{x^2}=1- \frac{4}{x^2}\\\\ 1- \frac{4}{x^2}=0\\\\ \frac{x^2-4}{x^2}=0\\\\ x\neq0\\\\ x^2-4=0\\ x^2=4\\ x=\pm2\\\\ f(2)=2+ \frac{4}{2}=4\\\\ f(1)=1+4=5\\\\ f(4)=4+1=5](https://tex.z-dn.net/?f=f%27%3D%28x%2B+%5Cfrac%7B4%7D%7Bx%7D%29%27%3Dx%27%2B+%5Cfrac%7B4%27%5Ccdot+x-4%5Ccdot+x%27%7D%7Bx%5E2%7D%3D1-+%5Cfrac%7B4%7D%7Bx%5E2%7D%5C%5C%5C%5C%0A1-+%5Cfrac%7B4%7D%7Bx%5E2%7D%3D0%5C%5C%5C%5C%0A+%5Cfrac%7Bx%5E2-4%7D%7Bx%5E2%7D%3D0%5C%5C%5C%5C%0Ax%5Cneq0%5C%5C%5C%5C%0Ax%5E2-4%3D0%5C%5C%0Ax%5E2%3D4%5C%5C%0Ax%3D%5Cpm2%5C%5C%5C%5C%0Af%282%29%3D2%2B+%5Cfrac%7B4%7D%7B2%7D%3D4%5C%5C%5C%5C%0Af%281%29%3D1%2B4%3D5%5C%5C%5C%5C%0Af%284%29%3D4%2B1%3D5+++++)
Ответ:
![\max [-1;4] \ f(x)=f(1)=f(4)=5\\ \min [-1;4] \ f(x)=f(2)=4](https://tex.z-dn.net/?f=%5Cmax+%5B-1%3B4%5D+%5C+f%28x%29%3Df%281%29%3Df%284%29%3D5%5C%5C%0A%5Cmin+%5B-1%3B4%5D+%5C+f%28x%29%3Df%282%29%3D4)
-log5 4x+log5 (x²+75)=1
log5 (x²+75)/4x=log5 5
x²+75)/4x=5 x²-20x+75=0 привели к общемузнаменателю 4х≠0
D=20²-4*1*75=400-300=100 x1=20+√100)/2=30/2=15 x2=20-10)/2=5
ответ 5 15