Question

а) √32-√128sin²(9π/8) найдите значение выражния
б) y=5x-ln(x+3)⁵+1 найдите минимум функции
пожалуйста с решением

Answer 1

4√2-8√2*(1-cos9π/4)/2=4√2-4√2(1-cos(2π+π/4)=4√2-4√2+4√2*cosπ/4=4√2*1/√2=4

y=5x-ln(x+3)^5 +1
y`=5-5(x+3)^4/(x+3)=5-5(x+3)³=0
5(x+3)³=5
(x+3)³=1
x+3=1
x=-2
y(-2)=-10-ln1+1=-10-0+1=-9

Answer 2

$y=5x-ln(x+3)^5+1 \\ y'=5- \frac{5(x+3)^4}{(x+3)^5}=5- \frac{5}{x+3} \\ x\ \textgreater \ -3 \\ y'=0 \iff 5- \frac{5}{x+3}=0 \iff 5(x+3)=5 \\ |:5 \\ x+3=1 \iff x=-2 \\ f(x)=y \\ f(-2)=5*(-2)-ln(-2+3)^5+1=-10-ln1+1=-9 \\ y_{MIN}=-9;x=-2$