Find all values of parameter $a$, that $ x^3-ax^2+3x-2=0$ has no solutions when $0<x<2$.

Question

Find all values of parameter $a$, that $f(x) = x^3-ax^2+3x-2=0$ has no solutions when $0<x<2$. I tried to find all zeros of derivative to find max and min value of $f(x)$ to check is $\min<0<\max$ and then substitute them in the equation, but then I failed. What should I do, or what theorem I should use?

Answer 1

$$f(x)={x^3+3x-2\over x^2}$$ is strictly increasing for $x>0$, so the line $y=a$ doesn't cuts the graph of $f$ for $x\in(0,2)$ if $a>f(2) = 3$.