For the curve y = 4x3 – 2x5 find all points

[Dhanalakshmi]

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

[Dhanalakshmi]

Let ( x₁, y₁) be the required point on y = 4x³ – 2x⁵. Then,

The equation of the given curve is y = 4x³ – 2x⁵. Differentiating w.r.t. x , we get

So, the equation of the tangent at ( x₁, y₁) is

This passes through the origin. Therefore,

Subtracting ( ii ) from ( i ) we get,

When x₁ = 0 ⇒ y₁ = 0 [Using ( ii )]
When x₁= 1 ⇒ y₁ = 12 – 10 = 2 [Using ( ii )]
When x₁= –1 ⇒ y₁= – 12 + 10 = – 2 [Using ( ii )]
Hence the required points are (0, 0), (1, 2) and (–1, –2).