1.
f(x)=ax^2+bx+k
f'(x)=2ax+b
f'(0)=b=0
f(x)=ax^2+k
f(1)=a+k
过(1,a+k)的切线斜率k1=f'(1)=2a+b=2a
x+2y+1=0的斜率k2=-1/2
所以k1*k2=(-1/2)(2a)=-a=1
a=-1
所以a=-1,b=0;
2.
g(x)=e^x/f(x)
=e^x/(-x^2+k)
g'(x)=[(-x^2+k)e^x-(-2x)e^x]/(-x^2+k)^2
=(-x^2+2x+k)e^x/(-x^2+k)^2
因e^x/(-x^2+k)^2＞0
所以g'(x)的正负与-x^2+2x+k相同,
-x^2+2x+k=-(x-1)^2+k+1
-(x-1)^2+k+1＞0时,即1-√(k+1)＜x＜1+√(k+1)时,g'(x)＞0,g(x)单调递增；
-(x-1)^2+k+1＜0时,即x＞1+√(k+1)或x＜1-√(k+1)时,g'(x)＜0,g(x)单调递减.