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 only way I know how to go about this problem is to find the derivative and thats it. Please Help!
It depends on how many different roots $\displaystyle x_0 \in \mathbb{R}$ the equation $\displaystyle f(x) = 0$ has.
Observe that between any 2 different roots $\displaystyle x_1,x_2$ that $\displaystyle f(x)$ attains a maximum/minimum in the interval $\displaystyle (x_1,x_2)$.
And there exists a $\displaystyle a_0$ with $\displaystyle x_1<a_0<x_2$ where $\displaystyle f'(a_0)(x-a_0) +f(a_0)$ the tangent at $\displaystyle (a_0,f(a_0))$ passes through the origin.
But we might just as well calculate how many maxima/minima f(x) attains:
Thus if you find all different "real " roots of $\displaystyle f'(x) = 20x^4-12x^3+30x = x(20x^3-12x^2+30) = 0$ you're done.
Edit: You don't even have to find the roots of $\displaystyle f'(x)= 0$ explicitly. You can use the intermediate value theorem to decide how many roots in $\displaystyle \mathbb{R}$ this equation has.