Question

(1 point) A function X(t) satisfies the differential equation di = (x − 2)(x – 5)²(x – 7)(x – 9)?. Compute the following limits. You can use words like “Infinity” and “DNE” if you need to. If x(0) = 3.5, then lim x(t) = DNE help (numbers) too If æ(0) = 6, then lim x(t) = DNE help (numbers) t- o If x(0) = 8, then lim x(t) = DNE help (numbers) to

Answer 1

1. dx = (x-2) (2-5² (x-7)(x-9) 2 dz Find critical point of alt) Put - +(x-2)(x-5) 2 (x-7)(x-9) 2-0 x=2, 5, 7, 9 of x42 . Then axo at - » xt) is increasing in ac2 If Then 2<x<5 dx co dt setts is decreasing in 24x<5 5< x < Then ad co dt 54&L 7 of > = xt) is decreasing in Tag > X(t) is increasing in 7cxLg

т т xtt) is increasing in x> 9. +) Now x = 9 A(0,8 ) x = 7 & B (0,6) x = 5 ac (0,3.5 x=2 Now As we 3.5 and we If x (0)=305 can see lies between 2cx<5 know in dax<5 dx co at decrear X(t) n 22x2 5 = x(t) > 2 lim too x tt) = 2 7 40) = 6 As we can see 6 lies between 5 and 7 a(t) is die rensing in 52 x 47 and sounded in 5<x<T

alt) - 5 lim too 200) = 6 we can see 8 lies between 7 and a and we know X(t) is increasing in and Bounded by 7 and 9 Tag g. If x(0) = 8 dim alt) = to