Question

At how many points on the curve `y = 4x^5-3x^4+15x^2+6` will the line tangent to the curve pass through the origin?

Answer 1

Any line that passes through the origin will be in the form `y=mx`, where `m` is some constant that is the slope of the line.

The slope of that line at any of the points we are looking for, which are tangent to the function, will have a slope equal to the function's derivative.

So, we want to find the times instead of `y=mx`, when `y=dy/dx(x)`.

Differentiating `y` through the power rule gives `dy/dx=20x^4-12x^3+30x`. Substituting this into `y=dy/dxx`, we get:

`4x^5-3x^4+15x^2+6=x(20x^4-12x^3+30x)`

`4x^5-3x^4+15x^2+6=20x^5-12x^4+30x^2`

`16x^5+9x^4-15x^2+6=0`

Graph to see how many solutions there are:

graph{16x^5+9x^4-15x^2+6 [-11.08, 14.24, -2.68, 9.98]}

There is only one time when this is the case.