Contact between curves

A contact between curves (point of contact) is a common point of two function graphs, where both functions have the same tangent (same slope).

Example

Determine the point of contact of the function $f(x)=x^2$ and $g(x)=-x^2+4x-2$.

1. Take derivatives

   $f(x)=x^2$
   $f'(x)=2x$
   
   $g(x)=-x^2+4x-2$
   $g'(x)=-2x+4$

2. Equate function equations

   First condition: Both functions must have a common point.
   $f(x_C)=g(x_C)$
   $x^2=-x^2+4x-2\quad|-x^2$
   $-2x^2+4x-2=0\quad|:(-2)$
   $x^2-2x+1=0$
   
   There is a quadratic equation that can be solved, for example, with the PQ formula.
   $x_{C_{1,2}} = -\frac{p}{2} \pm\sqrt{(\frac{p}{2})^2-q}$
   $x_{C_{1,2}} = 1 \pm\sqrt{1-1}$
   $x_C=\color{red}{1}$

3. Check slope

   Second Condition: Both functions must have the same slope at the point.
   $f'(x_C)=g'(x_C)$
   $f'(\color{red}{1})=g'(\color{red}{1})$
   $2\cdot\color{red}{1}=-2\cdot\color{red}{1}+4$
   $2=2$
   => The functions touch at the position $x_C=1$

4. Specify point of contact

   The point of contact should be specified: Therefore, calculate the y-coordinate with one of the original functions.
   
   $f(\color{red}{1})=\color{red}{1}^2=\color{blue}{1}$
   => Point of contact: $C(\color{red}{1}|\color{blue}{1})$