For the curve `y=4x^3-2x^5,`find all the points at which the tangent passes through the origin.

Explanation

Equation of curve
` y = 4x^(3) - 2x^(5)` ….(1)
` rArr (dy)/(dx) = 12x^(2) - 10x^(4)`
Slope of tangent at point `(x_(1), y_(1)) = 12x_(1)^(2) - 10x_(1)^(4)`
and equation of tangent.
` y - y_(1) = (12x_(1)^(2)-10x_(1)^(4))(x-x_(1))`
This tangent passes through the point (0, 0).
`:. 0-y_(1)=(12x_(1)^(2)-10x_(1)^(4))(0-x_(1))`
`rArr y_(1) = 12x_(1)^(3) - 10x_(1)^(5)` ....(2)`
`(x_(1), y_(1))` line on curve (1)
` :. y_(1)= 2x_(1)^(5)`
`rArr 12x_(1)^(3)-10x_(1)^(5)=4x_(1)^(3)-2x_(1)^(5)`
`rArr 8x_(1)^(3)-8x_(1)^(5)= 0`
`rArr 8x_(1)^(3) (1-x_(1)^(2))= 0`
` rArr x_(1) = 0, 1, -1`
From equation (2), the corresponding values of `y_(1)` are 0, 2 and -2.
`:. " Required point " -=(0, 0), (1, 2) and (-1, 2) and (-1, -2)`.