34. Determine the coefficients of the quadratic trinomial $a x^{2}+$ $+b x+c$, knowing that its maximum value is 25 at $x=\frac{1}{2}$ and that the sum of the cubes of its roots is 19.
34. $a=-4, b=4, c=24$.

Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.

34. $a=-4, b=4, c=24$.