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?

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.

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.

To find the points on the curve where the tangent line passes through the origin, we need to find the values of x that satisfy two conditions:

1. The y-coordinate of the curve at that point is zero (since the line passes through the origin).
2. The derivative of the curve at that point is equal to the slope of the line (which is zero since it is horizontal).

First, we find the derivative of the curve: y' = 20x^4 - 12x^3 + 30x.

Next, we set the derivative equal to zero and solve for x: 20x^4 - 12x^3 + 30x = 0.

Factoring out x, we get: x(20x^3 - 12x^2 + 30) = 0.

Setting each factor equal to zero, we have two possibilities:

1. x = 0
2. 20x^3 - 12x^2 + 30 = 0

Solving the second equation is more complex and requires numerical methods or approximations. However, we can determine the number of real solutions by analyzing the behavior of the cubic polynomial.

By observing the coefficients, we can see that the cubic term has a positive coefficient (20), indicating that the polynomial increases without bound as x approaches positive or negative infinity. Additionally, the constant term (30) is positive, indicating that the polynomial intersects the x-axis at least once.

Therefore, the cubic polynomial has at least one real root, and combined with the x = 0 solution, there are at least two points on the curve where the tangent line passes through the origin.