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Mathematics-Online problems:

Solution to the problem of the (previous) week


Problem:

The figure shows the graph of the polynomial

$\displaystyle p(x)=x^2(a-x^2)
$

as well as the tangent lines drawn at the points $ Q_i$ on the curve and thus forming a square.

\includegraphics[width=8.3cm]{TdM_05_A3_bild1.eps}


Answer:

$ P$ = $ \Big($ $ \sqrt a$, $ a^2 \Big)$
$ a$ =      
$ Q_1$ = $ \Big($ , $ \Big)$
$ Q_2$ = $ \Big($ , $ \Big)$
$ A$ =      

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


Solution:

From ALT= we obtain

ALT=

and for ALT= the graph of ALT= has the following form:
ALT=
The polynomial ALT= has exactly four points where the slope of the tangent lines is ALT= if the extrema of ALT= have ordinate values of ALT= Thus, ALT= has to be an inflection point. Therefore, we have

ALT=

Substituting ALT= we get

ALT=

i.e. ALT= and ALT=

For the abscissa of ALT= we have

ALT=

By symmetry, ALT= is one solution. We obtain the remaining solutions by applying polynomial division

ALT=

and using the formula for the roots of a quadratic equation:

ALT=

Hence, ALT=

In order to determine the area of the square, we first have to find the points where the tangent lines intersect the ALT=-axis:

ALT=

Using the length ALT= of the diagonal, we obtain the area

ALT=


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