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Mathematics-Online problems:

# Solution to the problem of the (previous) week

Problem:

The figure shows the graph of the polynomial

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

• Find the extremum of the polynomial in terms of the parameter and sketch the graph of the derivative for .
• For which do the tangent lines, whose slope is touch the polynomial at exactly points What is the area of the square enclosed by these tangents?

 = , = = , = , =

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

Solution:

From we obtain

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

Substituting we get

i.e. and

For the abscissa of we have

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

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

Hence,

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

Using the length of the diagonal, we obtain the area

[problem of the week]