I have two proofs here one which I did and the other was given in book. Is one better than the other? I am asking for in an exam setting which proof makes a better solution(as in fetch more marks).
Proof 1 (My proof)
$f'(x)=3x^2-3$, in the interval $(0,1)$ is less than $0$. If there were two distinct roots then $f'(0)$ should have been $0$ once in $(0,1)$ by rolle's theorem. Since it isn't there are no values of $k$ for which there are two real roots.
Proof 2 (Book)
Let $a,b$ be two roots of $f(x)$ in $(0,1)$ then there exists a $c$ such the $f'(c) = 0$ for c in $[a,b] $ by Rolle's theorem. $f'(c)= 3c^2-3$ has no solutions in $(0,1)$ hence there is no such value of $k$.
Best Answer
Both solutions are equally valid and should both get full marks.
Personally, I found your solution easier to follow though, but in an exam situation as long as what you’re doing is clear, a correct proof will achieve full marks regardless.