document.write( "Question 152664: This cubic, c(x) = x^3 – 2x^2 – 24x, has zeroes at x ∈ {-2, 0, 6}. What is the approximate value of its local minimum? @ As an approximation, we treat cubics, and other higher-order polynomials, as quadratics between their zeroes. Then, recognizing that the local minimum is between the second two zeroes, the local minimum is found at the midpoint between these zeroes.
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Algebra.Com's Answer #112225 by Edwin McCravy(20055) $\"\"$ $\"About$

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This cubic, c(x) = x^3 – 2x^2 – 24x, has zeroes at x ∈ {-4, 0, 6}. What is the approximate value of its local minimum? @ As an approximation, we treat cubics, and other higher-order polynomials, as quadratics between their zeroes. Then, recognizing that the local minimum is between the second two zeroes, the local minimum is found at the midpoint between these zeroes.
\n" ); document.write( "a. -63
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\n" ); document.write( "d. no way to know this
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document.write( "We draw the graph\r\n" );
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document.write( "The local minimum looks to be where the litle plus sign is:\r\n" );
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document.write( "We can't find it exactly with algebra (although it can be with\r\n" );
document.write( "calculus, as you will learn if you take calculus), but in\r\n" );
document.write( "algebra, we can approximate it by assuming the curve is like\r\n" );
document.write( "the green parabola below. We can see it's not exactly the same,\r\n" );
document.write( "but it will come fairly close.\r\n" );
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document.write( "We know that the x-coordinate of the vertex of the green parabola\r\n" );
document.write( "is half-way between its x-intercepts at 0 and 6. \r\n" );
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document.write( "Halfway between the x-intercepts (0,0,) and (6,0) is the point\r\n" );
document.write( "(3,0), so we will substitute x=3 into the equation\r\n" );
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document.write( "So the approximation of the local minimum, which is the vertex of\r\n" );
document.write( "the closely fitting green parabola, (3,-63), which is the second\r\n" );
document.write( "point marked below.\r\n" );
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document.write( "So the correct answer is a. -63.\r\n" );
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document.write( "As you see the two points are not the same, but that's as good as\r\n" );
document.write( "we can do with just algebra, and no calculus. \r\n" );
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document.write( "Edwin

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