Proving bounds on a and b for positive roots of ax³−x²+bx−1=0

[ritwik06]

Homework Statement

Suppose 'a' and 'b' are real numbers such that the roots of the cubic
equation ax^3-x^2+bx-1=0 are all positive real numbers. Prove that:
i) 0<3ab<=1
ii) b>= 3^0.5

Homework Equations

Let x,y,z be the roots:
x+y+z=1/a
xy+yz+zx=b/a
xyz=1/a

The Attempt at a Solution

I differentiated the above function. For the function to have three
distinct roots. the differentiated function (quadratic) should have 2
distinct roots. I put the discriminant >=0 to get part i.

But I cannot understand what shall I do with part ii?

I also noticed that the graph of the equation at x=0 is -1.
Help me further!

 

[HallsofIvy]

Actually, nothing is said about the three roots being distinct- only that then are positive real numbers.

 

[ritwik06]

Actually, nothing is said about the three roots being distinct- only that then are positive real numbers.

So Did I do the first on by the wrong method??