Mathematico | Mathematico

A curve is given by the equation

y = \frac{x^{2} - a}{x - a}, x \neq a

Find the range of values of $a$ for which the curve has exactly two stationary points.
Assuming $a$ is chosen from within this range, find, in terms of $a,$ the $x$ coordinate of each stationary point.
Hence, prove, using the second derivative test, that one of these stationary points is a maximum point and the other is a minimum point.

After finding $\frac{d y}{d x}$ using the quotient rule, you should find that the $x$ coordinates of the stationary points satisfy

x^{2} - 2 a x + a = 0

You can solve this by completing the square or by using the formula.

When you find the second derivative, it might look unpleasant, and you really aren't looking forward to substituting the $x$ coordinates of the turning points in. But look closely and you will see the quadratic $x^{2} - 2 a x + a$ again! The $x$ coordinates of the turning points are the roots of this quadratic (see the first hint), so we know this is going to become $0$ - this will save lots of algebra.