I found some really interesting graphs like these:
So basically through these graphs I want to ask that if we take 2 functions in the format of$$f_1 = ln(f(x))$$ and $$f_2 = \frac{f(x)}{e}$$ do they always intersect tangentially?
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$\begingroup$Well, for two functions to touch tangently in $x_0$, 2 conditions must meet (or 3 if you want to avoid intersection, but I'm not going to include this case):
- $$f_1(x_0) = f_2(x_0)$$
- $$f_1'(x_0) = f_2'(x_0)$$
If you substitute your functions in question to the conditions, we get:$$\ln(f(x_0)) = f(x_0) / e$$$$\frac{f'(x_0)}{f(x_0)} = f'(x_0) / e \implies f(x_0) = e ~\text{or}~f'(x_0)=0$$
Now if there exists $x_o\in\mathbb{R}$ s.t. $f(x_0) = e$ it follows that the first equality also holds,so you have a touching point, but not for a general $f$. For example $f(x)=1$
Edit:For the third condition mentioned above check out @Martin R's answer. He proves that $f_1(x) \le f_2(x) (\forall x \in \mathbb R)$. So the curves not only share the same tangent line at points where $f(x)=e$ but they are also "just touching" and not intersecting.
$\endgroup$ 2 $\begingroup$The inequality $\ln (u) \ge u/e$ holds for all $u > 0$, because the logarithm is concave, so that its graph lies below the tangent at $u=e$:$$ \ln (u) \le \ln (e) + (u-e)\frac 1e = \frac ue \, , $$with equality if and only if $u=e$.
Therefore, if $f(x) > 0$ then$$ h(x) = f_1(x) - f_2(x) = \ln f(x) -\frac{f(x)}{e} \le 0 \, , $$so that the graph of $f_1$ is never above the graph of $f_2$.
If the graphs “meet” at a point $x^*$ then $h$ has a maximum at that point. If $f$ is differentiable at $x^*$ then $h'(x^*) = 0$ follows, i.e. $f_1'(x^*) = f_2'(x^*)$.
That explains why $f_1$ and $f_2$ “meet tangentially”: they have the same slope at the intersection points.
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