A-A+
设f(x)和g(x)在(－∞ +∞)内都可导 且有fˊ(x)＞gˊ(x) f(a)＝g(a)
问题详情
设f(x)和g(x)在(－∞，+∞)内都可导，且有fˊ(x)＞gˊ(x)，f(a)＝g(a)，证明：当x＞a时，f(x)＞g(x)；当x＜a时，f(x)＜g(x)．
请帮忙给出正确答案和分析，谢谢！
参考答案
正确答案：证： 令 F(x)＝f(x)－g(x)则 F(a)＝f(a)－g(a)＝0又 Fˊ(x)＝fˊ(x)－gˊ(x)＞0所以函数F(x)在(－∞+∞)内单调增加从而在x＞a时F(x)＞F(a)即f(x)－g(x)＞0f(x)＞g(x)而x＜a时F(x)＜F(a)即f(x)－g(x)＜0f(x)＜g(x)证毕．
证：令F(x)＝f(x)－g(x),则F(a)＝f(a)－g(a)＝0,又Fˊ(x)＝fˊ(x)－gˊ(x)＞0,所以函数F(x)在(－∞,+∞)内单调增加,从而在x＞a时,F(x)＞F(a),即f(x)－g(x)＞0,f(x)＞g(x),而x＜a时,F(x)＜F(a),即f(x)－g(x)＜0,f(x)＜g(x),证毕．
