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Solution 2.3:10b

From Förberedande kurs i matematik 1

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The inequality y1x2 defines the area under and on the curve y=1x2, which is a parabola with a maximum at (0,1). We can rewrite the other inequality x2y3 as yx2+32 and it defines the area under and on the straight line y=x2+32.


 
The region y ≤ 1 - x² The region x ≥ 2y - 3


Of the figures above, it seems that the region associated with the parabola lies completely under the line y=x2+32 and this means that the area under the parabola satisfies both inequalities.


The region y ≤ 1 - x² and x ≥ 2y - 3


Note: If you feel unsure about whether the parabola really does lie under the line, i.e. that it just happens to look as though it does, we can investigate if the y-values on the line yline=x2+32 is always larger than the corresponding y-value on the parabola yparabola=1x2 by studying the difference between them

ylineyparabola=x2+23(1x2).

If this difference is positive regardless of how x is chosen, then we know that the line's y-value is always greater than the parabola's y-value. After a little simplification and completing the square, we have

ylineyparabola=x2+23(1x2)=x2+21x+21=x+412412+21=x+412+716

and this expression is always positive because 716 is a positive number and x+412  is a quadratic which is never negative. In other words, the parabola is completely under the line.