f(x)=kx^3-3(k+1)x^2-k^2+1(k大于0)1）若f(x)的单调减区间为（0,4）求k值2）当x大于1时,求证2x^(1/2)大于3-(1/x)

Question

f(x)=kx^3-3(k+1)x^2-k^2+1 (k大于0)
1）若f(x)的单调减区间为（0,4）求k值
2）当x大于1时,求证2x^(1/2) 大于 3-(1/x)

Answer 1

1 f'(x)=3*k*x^2-6*(k+1)x < 0
x1=0
x2=2(k+1)/k=4
所以 k=1
2 记 f(x)=2x^(1/2)+x^(-1)-3=2x^(1/2)+1/x-3, 则f'(x)=x^(-1/2)-x^(-2)=
(x^(3/2)-1)/x^2. 因为 x>1 时x^(3/2)-1>0, 从而f'(x)>0,f(x)单调递增,
而 x<1 时,x^(3/2)-1<0, 从而f'(x)<0,f(x)单调递减,所以函数f(x)在x=1
时取得最小值,因为f(1)=0,即最小值等于0,所以f(x)在 x>0 时的值不小于0,
即有 2x^(1/2)+x^(-1)-3>=0, 所以 2x^(1/2)>=3-x^(-1)