Equation of the curve is y=x3−6x2+11x−6
This equation can be written as,
y=(x−1)(x−2)(x−3)
When x=0, y=-6.
So, I can directly find the x where y=0,
It will give me, x=1,2,3
So I will have intervals as (−∞,1],[1,2],[2,3],[3,∞)
In interval (−∞,1], If I put any value from −∞ to 1, value of the y will increase.
In interval [3,∞) , on increasing value of x, y will increase.
Now, dxdy=3x2−12x+11
Putting dxdy=0, We get, x=1.423,2.577
dx2d2y=6x−12
At x=1.423, dx2d2y=6(1.423)−12<0, Hence it is point of maxima.
At x=2.577, dx2d2y=6x−12>0 , So it is point of minimum.
Using all these properties, We find that curve will look like,
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