MHB Differentiation / Integration Help

[Joe20]

The curve has a gradient function dy/dx = 2 +q/(5x^2) where q is a constant, and a turning point at (0.5, -4). Find the value of q.

option 1 : 2.5
option 2: -2.5
option 3: 0
Option 4: -3

I couldn't find the answer and will need assistance to how the answer can be obtained.

I have substituted x = 0.5 into dy/dx to get the gradient expression of 2 + 4q/5 and integrated to get y = 2x - q/(5x) + c.
It seems impossible for me to get the value of q since c could not be found. I am not sure if the question has some missing information to continue. Your help will be greatly appreciated. Thanks.

 

[MarkFL]

Since the given point is a turning point, we must have:

$$\left.\d{y}{x}\right|_{x=\frac{1}{2}}=2+\frac{q}{5\left(\frac{1}{2}\right)^2}=2+\frac{4q}{5}=0$$

Now you just need to solve for $q$.