# Show there can't be two real and distinct roots of polynomial $f(x)=x^3-3x+k$ in $(0,1)$, for any value of k.

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 $$k$$.

• Yours is better because the book version has a wrong expression for $f'$ – Hagen von Eitzen Jun 11 at 6:09
• – lab bhattacharjee Jun 11 at 6:16
• @HagenvonEitzen that's a typo sorry. – Sonal_sqrt Jun 11 at 6:26

## 1 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.