Question

determine the position and nature of the double points on the curve to X ki power 4 minus y cube minus 12 y square - 4 x square - 2 equals to zero​

Answer 1

The position and nature of the double points on the curve to X ki power 4 minus y cube minus 12 y square - 4 x square - 2 equals to zero​

Step-by-step explanation:

1. df/dx = 8x2-8x  
2. df/dy = -24y2-24y

d2f/dx2 = 24x2-8

d2f/dy2 = -48y-24

Now need to calculate of the position of double point on the curve

df/dx = 0

8x3-8x = 0

8x2-8 = 0

x = (1,0)

To use y as first derivative:

df/dy = 0

-24y2-24y = 0

-24y-24 = 0

y = (-1,0)

Now use second derivative x and y

d2f/dx2 = 24x2-8

now if x = 0 then:

d2f/dx2 = -8

The point maximum is 0

If we put x = 1 then:

d2f/dx2 = 16

The point minimum is = 1

Now for the derivative y is

d2f/dx2 = -48y2-24

now putting value of y = 0

d2y/dx2 = -24

The point maximum will y = 0

If we put y = 1

then d2f/dx2 = 24

the point minimum for x = 1

Hence it proved that the position of double point on the curve is to be 1,0 and -1,0

The nature will be x = 0  and x = 1

for y is y = 0 and y = 1

Learn more : Derivatives

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