# Thread: Tangent Lines Passing Through Origin

1. ## Tangent Lines Passing Through Origin

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.

2. 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.

3. If you continue by using $\displaystyle \frac{f(x)}{x}$

then you can find the minimum value(s) of this,
that will give you the slope of the tangent(s) that passes through the origin,
allowing for x being positive or negative.

4. In blue is f(x) and in pink is $\displaystyle \frac{f(x)}{x}$

the tangent slope gives the min value for $\displaystyle \frac{f(x)}{x}$

since if another x is chosen to the right of the origin, the ratio will be greater.