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We have two functions:

$$\begin{align*} y &= x^4-5x^3+2x^2-5 \\ y &= -11x-20 \end{align*}$$

My task is to find two closest points that can be found on these two functions.

Can somebody give any hints on how to solve these type of exercises?

Thank you!

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Do you mean the closest points of the curves, or the minimum difference between the two functions? –  copper.hat Feb 8 '13 at 8:17
The minimum difference between the two functions. :) –  Trom Feb 8 '13 at 8:23
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2 Answers

up vote 2 down vote accepted

You want to minimize $x^4-5 x^3+2 x^2+11 x+15$. Take the derivative, set that to $0$, and solve. Unfortunately this is an irreducible cubic, so the solutions are not very nice. There are three real roots: approximately $-.6828742578$, $1.275428188$, $3.157446070$. I'll leave it to you to find which of these gives the minimum.

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Since you want the minimum difference between $$\begin{align*} y_1(x) &= x^4-5x^3+2x^2-5 \text{ and }\\ y_2(x) &= -11x-20 \end{align*}$$ you are looking for the minimum of $$ \left| y_1(x) - y_2(x) \right|$$ which will be a max / min of $$ z(x) = y_1(x) - y_2(x).$$ So write down the equation for $z$ as Robert Israel has done for you, differentiate $z$ and find its turning points.

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