Question

二次函数f(x)=ax²+bx+c(a≠0),且Δ=b²-4ac,试判断下列命题的真假,若命题为假,则举出一个反例说明;若命题为真,则证明之.

命题(1):若Δ＜0,则af(x)≤0;

命题(2):若Δ＞0,x₁、x₂是方程f(x)=0的两根,且x₁＜x₂,则当x₁＜x＜x₂时,af(x)＜0;当x＜x₁或x＞x₂时,af(x)＞0.

Answer 1

解：(1)假命题.若a＞0,则af(x)＞0,如f(x)=x²+1.

(2)真命题.a＞0时,f(x)=a(x-x₁)(x-x₂).

当x₁＜x＜x₂时,f(x)＜0,af(x)＜0;

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

当x₁＜x＜x₂时,f(x)＞0,af(x)＜0;

当x＜x₁或x＞x₂时,f(x)＜0,af(x)＞0.