Let $f(x)=x^{4}-6x^{2}+5$. If $P(x_0,y_0)$ is a point such that $y_0>f(x_0)$ and there are exactly two distinct tangents through P

Question

Let $f(x)=x^{4}-6x^{2}+5$. If $P(x_0,y_0)$ is a point such that $y_0>f(x_0)$ and there are exactly two distinct tangents through P drawn to the curve $y=f(x)$, then find the maximum possible value of $y_0$.

The hint said P is the point of intersection of tangents drawn at inflection points.

I calculated inflection points, and they came out to be $(1,0), (-1,0)$.

I calculated tangents at these points and found the point of intersection to be $(0,8)$.

So, the answer is $8$, which is correct but the doubt is why this is happening this way. What's the importance of inflection point here?

Answer 1

I don't think the answer is correct.

For example the tangent to the curve at $x=-\frac32$ is $$r: y=\frac{9}2x + \frac{53}{16}$$ and the tangent line to the curve at $x=-\frac14$ is $$s : y=\frac{47}{16}x + \frac{1373}{256},$$ and they interesect at $P\left(\frac{21}{16},\frac{295}{32}\right)$.

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EDIT

The correct answer is the following. The supremum of $y_0$ is the only real positive $y$-solution to the system $$\begin{cases}y&=&x^4-6x^2+5\\ y&=&8x+8,\end{cases}$$ that is the ordinate of the interesction point between the given curve and the tangent line at the inflection point $(-1,0)$.

We get $y_{sup}=32$ (see dashed line in the figure below).

In order to draw the above conclusion you can use the following observations.

1. Any point $P$ whose ordinate is such that $y> 8|x|+8$ (that is any point above the the two tangent lines shown in the figure) can be connected to the concave part of the graph $f(x)=x^4-6x^2+5$ only with a straight line whose angular coefficient has absolute value greater than $8$, which shows that such line cannot be tangent to the curve.
2. On the other hand, consider any point $P_1$ on the curve, whose abscissa $x_{P_1}$ is, e.g., $-1<x_{P_1}<0$, and let $r_1$ be the line tangent to the curve at $P_1$. By appropriately selecting a second tangency point $P_2$ with abscissa $-3<x_{P_2}<-1$ we can find a second tangent line $r_2$ whose intersection with $r_1$ is any point $P\in r_1$ with $x_P>x_{P_1}$.