I have a cubic equation:
$y = 2x^3-12x^2+18x$
I want to search its extreme relative (extreme point). What should i do? With which process should i use?
And can I do this equation with that process?
$y = x^3+3x^2$
Note: I'm beginner
I have a cubic equation:
$y = 2x^3-12x^2+18x$
I want to search its extreme relative (extreme point). What should i do? With which process should i use?
And can I do this equation with that process?
$y = x^3+3x^2$
Note: I'm beginner
I intend to search for one of your functions, say the first one: $y = 2x^3-12x^2+18x$ The method says do the first derivative of $y$ and put it to zero because at relative extreme points $y'$ exists and is equal to $0$. You get $y'=6x^2-24x+18=0\longrightarrow 6(x-1)(x-3)=0$ and then $x=1,x=3$. These values can lead us to the relative extreme points, called critical points. Now you can use another method to find out which one of above $x$ is related to $max$ and which one is related to $min$. It says if some $x$ is caused by solving $y'=0$ such that $y'' $ exists, then if $y''(x)>0$ then that $x$ lead us to $min$ point and if $y''(x)<0$ then that $x$ lead us to $max$. Here, we have $y''=12x-24$ and so $x=1\longrightarrow y''\big|_{x=1}=12-24=-12<0$ so $y(1)=2-12+18=8$ is the relative maximum. And so $x=1\longrightarrow y''\big|_{x=3}=36-24=+12>0$ so $y(3)=0$ is the relative minimum. Here a plot of the first function: