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Find the lowest degree polynomial that satisfies the following constraints:

i) F(0)=0

ii)F(1)=0 $F(0)=0$

ii) $F(1)=0$

iii)The maximum of F $F$ on the interval (0,1) $(0,1)$ occurs at point c $c$

iv) F(x) $F(x)$ is positive on the interval (0,1) $(0,1)$

The answer seems to depend pretty strongly on c. $c$. It's not difficult to find solutions for all c, $c$, but the solutions are not minimal. It seems like the solution involves Chebyshev polynomials, but I'm not familiar with them. Can anyone recommended a link?
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Minimal polynomial with a given maximum in the unit interval

Find the lowest degree polynomial that satisfies the following constraints:

i) F(0)=0

ii)F(1)=0

iii)The maximum of F on the interval (0,1) occurs at point c

iv) F(x) is positive on the interval (0,1)

The answer seems to depend pretty strongly on c. It's not difficult to find solutions for all c, but the solutions are not minimal. It seems like the solution involves Chebyshev polynomials, but I'm not familiar with them. Can anyone recommended a link?