Question

Sketch the graph of y = x³ − 3x + 3.

Answer 1

At x-intercepts you must have y=0, hence we need to find the real root of y=0.

. Hence ​​​​​​the point (a,y(a)) is the x-intercept of the curve.where -1<a<-3.

At y intercepts, x=0 hence the point (0,3) is the​​​​ y-intercept.

At the extremum points you must have the derivative of y that is y'=0. Now, y'=3x²-3 &. y"=6x

So y' = 0 implies x = ±1.

at x=1, you will have y=1 and y"= 6 >0 so the point (1,1) is locally minimum.

at x=-1, you get y=5 and y"= -6 < 0, so the point (-1,5) is the point of locally maximum.

And as x± infinity you get, y± infinity.

And from the above information on (x,y) you can draw the curve roughly....