Finding stationary points without being able to solve for x?

Question

$ () = ^3 + 2x $
$\lambda $ is a real parameter.

Find its stationary point(s) and discuss the nature of the stationary point(s), in the case where $\lambda > 0$, $\lambda = 0$, and $\lambda < 0$.

The problem I am having is in the fact that it isn't possible to substitute $0$ in and solve for $x$ without getting an imaginary number? Thus you can't find the $x$ co-ordinates of the stationary points and then finish the rest of the question?

I.e

$ f'(x) = 3x^2 + 2\lambda $

No number goes into $x$

How are you supposed to complete the question if you can't find the $x$ co-ordinate?

I can only get $\sqrt{-\frac 23}$, which doesn’t help me?

Answer 1

The stationary points are thos points $x$ such that $f'(x)=0$. Therefore:

• there are no stationary points if $\lambda>0$;
• there is one and only one stationary point when $\lambda=0$, which is $0$;
• there are two and only two stationary points when $\lambda<0$, which are $\pm\sqrt{-\frac{2\lambda}3}$.