Find the stationary points

Question

Determine the stationary points of the following function and for each stationary point determine whether it is a local maximum, local minimum or a point of inflexion.

$f(x)=x^3(x-1)^2$

Using product rule to find $dy/dx$ I got:

$5x^4-8x^3+3x^2$ how can I factorize this and get the two $x$ values?

$5x^4-8x^3+3x^2=0$ how to find the two values from here?

Answer 1

Notice that, by factoring out $x^2$,

$$5x^4 - 8x^3 + 3x^2 = x^2(5x^2 - 8x + 3) = 0$$

Obviously, the left factor in the result gives you $x=0$ as a potential zero of $f'(x)$. You can use your method of choice, e.g. the quadratic formula, to find the zeroes of the quadratic factor and obtain your other zeroes for $f'(x)$.

Answer 2

$f'= 5x^4-8x^3+3x^2 =x^2 (5x-3)(x-1) $ to obtain the roots.

You can use second derivative to classify two of the solutions. However, at $x=0$ it's not suitable. The easiest way was simply sketch $f'$, i.e. the phase diagram like that was presented during the class.