Question

设函数f(x)=1/3ax^3+1/2bx^2+cx(a,b,c∈R,a≠0）的图像在[x,f(x)]处的切线的斜率为K(X)

设函数f(x)=1/3ax^3+1/2bx^2+cx (a,b,c∈R,a≠0）的图像在x,f(x)处的切线的斜率为k(X), 且函数g(X)=k(X)-X/2为偶函数
若函数k(X)满足下列条件：
1.k(-1)=0
2 对一切实数x,不等式k(X)≤x^2/2+1/2恒成立

（1）求函数K(X)的表达式

（2）求证1/k(1)+1/k(2)+·········+1/k(n)＞2n/n+2

Answer 1

(1)
k(x)=ax^2+bx+c
g(x)=ax^2+(b-1/2)x+c为偶函数, b-1/2=0 , b=1/2
k(x)=ax^2+x/2+c
k(-1)=a-1/2+c=0 a+c=1/2
ax^2+x/2+c≤x^2/2+1/2
判别式=1/4-4*(a-1/2)*(c-1/2)≤0
1/4-4ac+2(a+c)-1≤0
1/4≤4ac
(a+c)^2≤4ac
(a-c)^2≤0
a=c=1/4
k(x)=x^2/4+x/2+1/4
(2)
k(x)=(x+1)^2/4
S(n)=1/k(1)+1/k(2)+1/k(3)+....1/k(n)
=4/2^2+4/3^2+4/4^2+....+4/(n+1)^2
=4*[1/2^2+1/3^2+1/4^2+...+1/(n+1)^2]
>4*[1/(2*3)+1/(3*4)+1/(4*5)+...+1/(n+1)(n+2)]
=4*[1/2-1/3+1/3-1/4+1/4-1/5+...+1/(n+1)-1/(n+2)]
=4*[1/2-1/(n+2)]
=2n/(n+2)
S(n)>2n/(n+2)