The curve y = 4x^2 + a/ x + 5 has a stationary point. Find the value of the positive constant a given that y-ordinate of the stationary point is 32.
This question might look hard but can be easily answered by breaking it down into steps and remembering a few key things. The first step is to recognise that the question is mentioning a stationary point which means dy/dx = 0 this will allow us to solve for our unknowns in the equation y = 4x2 +ax-1 + 5 (1). A good practice is to write out the equation using the correct indices, this will make life easier when applying rules of indices later. First, we must differentiate Eq (1). dy / dx = 8x - ax-2 (2). As we are going to be using y at the stationary point to find a, a possible second step would be to find a in terms of x. setting dy / dx = 0 we get 8x - ax-2 = 0, ax-2 = 8x. Here it is really important to remember our rules of indices: a = 8 x1 / x-2 , a = 8 x(1--2) , a = 8x3 (3). So now we know a in terms of x at the stationary point. We can now substitute in Eq (3) into Eq (1) when y = 32. 4 x2 + (8x3)x-1 + 5 = 32, 4x2 + 8x2 + 5 = 32, 4x2 + 8x2 = 27 , 12x2 = 27, x2 = 27/12. A level questions will often have nice round numbers so it is worth noting here that 27 and 12 are both divisible by 3 so we can write x2 = 9/ 4, two nice square numbers gives x = 3/2 (4) . As a is a cube of x and the question asks for a positive value we can just use the positive root. We can now find a by substituting Eq (4), into Eq (3): a = 8 (3/2)3, a = 8 X 27 / 8 giving a = 27.
