Question

serenity

想請問這題

(1) ˊ 已知多項式(x)滿足x²f(x)+(x-1)f'(x)-9f(x)=0且f(0)=
1=3,則下列選項哪些正確?
3
(1). (x)為3次多項式 (2)y=f(x)有一個極大值及一個極小值(3)y=f(x)在x=-處有一
4 反曲點 (4) (2018)>f(2020).

Answer 1

imma_idiot

x²f'' +xf' -9f =f', 注意到次數少了 1 → let deg(f) =n,
n(n -1) +n -9 =0 (最高次消失), n =3 (負不合);
a₀ =f(0) =1/3,
x0: -9a₀ = a₁, a₁ =-3
x1: a₁ -9a₁ =2a₂, a₂ =12
x2: 2a₂ +2a₂ -9a₂ =3a₃, a₃ =-20
F =-20x³ +12x² -3x +1/3;

A. n =3 (v)
B. a₁a₃ >0 → 無極值 (x)
C. -b/3a =1/5 (v)
D. a₃ <0 (v)