Determine the points of intersection of the line tangent to $\frac{1}{x^2}$ on (a, $\frac{1}{a}$) with f(x).

Question

Determine the points of intersection of the line tangent to f(x) = $\frac{1}{x^2}$ on (a, $\frac{1}{a^2}$) with f(x).

So I took the derivative of f and ended up with the following equation: $\frac{-2}{a^3}x^3 + \frac{3}{a^2}x^2 -1 = 0 $

How do I take it from here? How do I solve the equation?

Answer 1

I agree with the equation you're trying to solve. It would be easier to deal with if you multiply through by $a^3$, so you're left with

$$-2x^3 + 3ax^2 -a^3.$$

We are looking for $x$ that satisfy this. In general cubics are not easy to solve. However, in this case we already know a root of this cubic: $x=a$. Therefore you can factor out $x-a$ and solve the remaining quadratic. You then need to check if the solutions that you have found are indeed solutions to your original problem. I hope this helps.