The curve y = 4x^2 + a/x +5 has a stationary point. Find the value of the positive constant 'a' given that the y-coordinate of the stationary point is 32. (OCR C1 2016)

First, we must find the value of x that corresponds to the stationary point, i.e. where the gradient (dy/dx) is zero. This will be in terms of 'a' (if we change 'a,' we change the shape of the function by a transformation that is not linear, which means we will change the x-coordinate and the y-coordinate corresponding to the stationary point). Differentiating (and remembering that a/x = ax^-1):
dy/dx = 8x -ax^-2 = 0

8x = ax^-2

8x³ = a

x = a^1/3/2

As mentioned, plugging in this value of x will always allow us to find the x-coordinate of the stationary point.Now, we must find the value of 'a' which makes the stationary point correspond to a y-value of 32 (using some quite involved algebra!):

32 = 4[(a^2/3)/4] + a/(a^1/3/2) + 5

27 = a^2/3+ 2a^2/3

9 = a^2/3

a = 9^3/2 = 27