Classifying points found with derivative

Question

I've been trying to learn derivatives recently. I have troubles with finding local minimums and maximums given an equation

I have understood this far

$y=2x^3-9x^2+12x-5$

$dy/dx= 6x^2-18x+12$

$x=1$ or $x=2$

$f(1)=0$
Point A $(1,0)$

$f(2)=-1$
Point A $(2,-1)$

What I do not understand is how one categorizes the points, either as minimums or maximums. I have looked around but I can't make sense of the explanations.

Answer 1

If:

• ${d^2 y \over d x^2} > 0:$ local minimum
• ${d^2 y \over d x^2} < 0:$ local maximum
• ${d^2 y \over d x^2} = 0:$ point of inflection

If the second derivative is positive, that means the first derivative (i.e., the slope) keeps getting larger at the point in question. The only way for that to be is for the point to be a local minimum.

Conversely if the second derivative is negative.