Showing a function does not have two distinct roots

Question

I have the function $f(x)=x^3-(3/2)x^2+k$ where $k$ is any real number and I am to show that the function does not have two distinct roots in the interval $[0,1]$.

I am in need of help applying the mean value theorem and rolle's theorem to prove this.

Thanks in advance.

Answer 1

Suppose towards a contradiction that $f$ has two distinct roots $a$ and $b$ in the unit interval (say $b>a$). Then by the mean value theorem $$0=f(b)-f(a)=f'(c)(b-a)=(3c^2-3c)(b-a)$$

for some $c\in(0,1)$. But does the derivative vanish in $(0,1)$?