ANSWER
[tex]x \: > \: 17 [/tex]
EXPLANATION
The given function is
[tex]f(x) = 2 {x}^{2} - 31x - 51[/tex]
Let us split the middle term to get,
[tex]f(x) = 2 {x}^{2} -34x + 3x- 51[/tex]
We factor to obtain,
[tex]f(x) = 2x(x - 17) -+3(x - 17)[/tex]
[tex]f(x) = (2x -+3)(x -17) [/tex]
Let us find the zeros of f(x),
[tex] (2x +3)(x - 17) = 0[/tex]
Either
[tex]2x +3 = 0 \: or \: x - 17 = 0[/tex]
[tex]x = -\frac{3}{2} \: or \: x = 17[/tex]
We need to find the x-value of the vertex, which is the midpoint of the x-intercepts.
This gives us,
[tex] \frac{- \frac{3}{2} + 17}{2} = 7.75[/tex]
Hence the axis of symmetry of the parabola has equation,
[tex]x = 7.75[/tex]
Since the given function has minimum turning point, it will decrease and then increase.
The function will be increasing for
[tex]x \: > \: 7.75[/tex]
The function will be increasing and positive for
[tex]x \: > \: 17[/tex]
