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

(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)