Question

二次函數f(x)=ax²+bx+c(a≠0),且Δ=b²-4ac,試判斷下列命題的真假,若命題為假,則舉出一個反例說明;若命題為真,則證明之.

命題(1):若Δ＜0,則af(x)≤0;

命題(2):若Δ＞0,x₁、x₂是方程f(x)=0的兩根,且x₁＜x₂,則當x₁＜x＜x₂時,af(x)＜0;當x＜x₁或x＞x₂時,af(x)＞0.

Answer 1

解：(1)假命題.若a＞0,則af(x)＞0,如f(x)=x²+1.

(2)真命題.a＞0時,f(x)=a(x-x₁)(x-x₂).

當x₁＜x＜x₂時,f(x)＜0,af(x)＜0;

當x＜x₁或x＞x₂時,f(x)＞0,af(x)＞0,a＜0時,f(x)=a(x-x₁)(x-x₂).

當x₁＜x＜x₂時,f(x)＞0,af(x)＜0;

當x＜x₁或x＞x₂時,f(x)＜0,af(x)＞0.