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Mathematics-Online problems:

Interactive Problem 615: Polynomials with Orthogonal and Parallel Tangents

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The polynomials $ p$ and $ q$ of degrees 2 and 4, respectively, are symmetric about the y-axis, touch each other at the point $ \left(0,\frac{1}{4} \right)$ and form a right angle at their point of intersection $ (1,0)$.

\includegraphics[width=.5\linewidth]{Polynom_Bild1}

Determine the coefficients of $ p$ and $ q,$ the zeros and minima of $ q,$ and the area $ A$ of the shaded region.


Answer:

$ p$ $ =$ $ -1/$$ x^2+1/$
$ q$ $ =$ $ 5/$$ x^4-3/$ $ x^2+1/$
Zeros: $ (\pm 1,0) ,\ (\pm$ $ ,0)$
Minima: $ (\pm$ $ ,$ $ )$
$ A$ $ =$ $ 1/$

(The results should be correct to two decimal places.)


   

(From: Day of Mathematics 2004)
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  automatically generated: 8/11/2017