By symmetry,
$R = \frac{1}{2}(15 \times 10)$
$R = 75 \, \text{kN}$
$M_\text{support} = -15x \left( \dfrac{x}{2} \right)$
$M_\text{support} = -7.5x^2$
$M_\text{midspan} = 75(5 - x) - 15(5)\left( \dfrac{5}{2} \right)$
$M_\text{midspan} = 75(5 - x) - 187.5$
$M_\text{midspan} = 187.5 - 75x$
If $x = 2 \, \text{m}$
$M_\text{midspan} = 187.5 -75(2)$
$M_\text{midspan} = 37.5 \, \text{kN}\cdot\text{m}$ ← Answer for Part (1)
For $M_\text{midspan} = 0$
$187.5 - 75x = 0$
$75x = 187.5$
$x = 2.5 \, \text{m}$ ← Answer for Part (2)
The least possible value of maximum moment will occur when $M_\text{support}$ is numerically equal to $M_\text{midspan}$
$| M_\text{support} | = | M_\text{midspan} |$
$7.5x^2 = 187.5 - 75x$
$7.5x^2 + 75x - 187.5 = 0$
$x = 2.071 \, \text{m}$
$L = 10 - 2x$
$L = 10 - 2(2.071)$
$L = 5.858 \, \text{m}$ ← Answer for Part (3)
