ANSWER
![x \: > \: 17](https://tex.z-dn.net/?f=x%20%5C%3A%20%3E%20%5C%3A%2017%20)
EXPLANATION
The given function is
![f(x) = 2 {x}^{2} - 31x - 51](https://tex.z-dn.net/?f=f%28x%29%20%3D%202%20%7Bx%7D%5E%7B2%7D%20-%2031x%20-%2051)
Let us split the middle term to get,
![f(x) = 2 {x}^{2} -34x + 3x- 51](https://tex.z-dn.net/?f=f%28x%29%20%3D%202%20%7Bx%7D%5E%7B2%7D%20-34x%20%2B%203x-%2051)
We factor to obtain,
![f(x) = 2x(x - 17) -+3(x - 17)](https://tex.z-dn.net/?f=f%28x%29%20%3D%202x%28x%20-%2017%29%20-%2B3%28x%20-%2017%29)
![f(x) = (2x -+3)(x -17)](https://tex.z-dn.net/?f=f%28x%29%20%3D%20%282x%20-%2B3%29%28x%20-17%29%20)
Let us find the zeros of f(x),
![(2x +3)(x - 17) = 0](https://tex.z-dn.net/?f=%20%282x%20%2B3%29%28x%20-%2017%29%20%3D%200)
Either
![2x +3 = 0 \: or \: x - 17 = 0](https://tex.z-dn.net/?f=2x%20%2B3%20%3D%200%20%5C%3A%20or%20%5C%3A%20x%20-%2017%20%3D%200)
![x = -\frac{3}{2} \: or \: x = 17](https://tex.z-dn.net/?f=x%20%3D%20-%5Cfrac%7B3%7D%7B2%7D%20%5C%3A%20or%20%5C%3A%20x%20%3D%2017)
We need to find the x-value of the vertex, which is the midpoint of the x-intercepts.
This gives us,
![\frac{- \frac{3}{2} + 17}{2} = 7.75](https://tex.z-dn.net/?f=%20%5Cfrac%7B-%20%5Cfrac%7B3%7D%7B2%7D%20%2B%2017%7D%7B2%7D%20%3D%207.75)
Hence the axis of symmetry of the parabola has equation,
![x = 7.75](https://tex.z-dn.net/?f=x%20%3D%207.75)
Since the given function has minimum turning point, it will decrease and then increase.
The function will be increasing for
![x \: > \: 7.75](https://tex.z-dn.net/?f=x%20%5C%3A%20%3E%20%5C%3A%207.75)
The function will be increasing and positive for