Smallest distance between two functions

[Math Tutor]

What is the smallest distance between these two functions:
f(x) = -3x - 4
and
f(x) = -20x^2 + 4x + 1

[razemsoft21]

teacher wrote:
> What is the smallest distance between these two functions:
> f(x) = -3x - 4
> and
> f(x) = -20x^2 + 4x + 1

The normal to f(x) = -3x - 4 is -3

Find the tangent to f(x) = -20x^2 + 4x + 1
f '(x) = -40x + 4 =-3 ---> x = 7/40
find y when x = 7 / 40 ---> y = -20*49/1600 + 4*7/40 + 1

Finaly find the smallest distance between f(x) = -3x -4
and the point (7/40 , -20*49/1600 + 4*7/40 + 1) using:

[tex]d=\frac{|ax1 + by1 + c|}{\sqrt{a^2 + b^2}}[/tex]

[Math Tutor]

Thank you for the perfect solution!
Tomorrow I will write the next problem of the month.

[Guest]

Hello,
Could you show example for higher order functions?
Let g and f be cubic polynomials.
f(x) = ax^3 + bx^2 + cx + d
g(x) = ex^3 + fx^2 + gx + h

I have tried and it went so messy, that two intersecting polynomials went complex :-/

Is it possible to do the same calculations that way?
Would it be possible to derive formula for polynomials of two variables?

Best Regards,
Maciek