# For the curve y = 4x3 – 2x5, find all the points at which the tangent passes through the origin.

The given curve y = 4x3 – 2x5

Then, the slope of the tangent at the point (x, y) is 12x2 – 10x4

The equation of the tangent at (x,y) is given by,

Y – y = (12x2 – 10x4)(X – x) ………….(1)

When the tangent passes through the origin (0,0), then X =Y=0

Then equation (1) becomes,

-y = (12x2 – 10x4)(– x)

y = (12x3 – 10x5)

Also, we have y = 4x3 – 2x5

(12x3 – 10x5) = 4x3 – 2x5

8x5 – 8x3 = 0

x5 – 2x3 = 0

x3(x2 – 1) = 0

x = 0 , 1

When x = 0 then y = 0

When x = 1 then y = 2

And when x = -1 then y = -2

Therefore, the required points are (0, 0), (1,2) and (-1, -2).

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