document.write( "Question 1036690: Given the function $\"+f%28x%29+=+x%5E3-3x%2B2+\"$ on the domain $\"+a%3C=x%3C=a%2B2\"$ , find the value of a at which two ends of the domain give the same value , equal to the maximum value of the interval. \n" ); document.write( "

Algebra.Com's Answer #651472 by rothauserc(4718) $\"\"$ $\"About$

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The graph of the equation f(x) = x^3 -3x + 2 is
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\n" ); document.write( " $\"+graph%28+300%2C+200%2C+-2.5%2C+2.5%2C+-5%2C+10%2C+x%5E3+-3x+%2B+2%29+\"$
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\n" ); document.write( "note that x^3 -3x +2 = (x-1)^2 * (x+2)
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\n" ); document.write( "we are looking for the value of a that satisfies
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\n" ); document.write( "a^3 -3a + 2 = (a+2-1)^2 * (a+2+2)
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\n" ); document.write( "a^3 -3a +2 = a^3 +6a^2 +9a +4
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\n" ); document.write( "6a^2 +12a +2 = 0
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\n" ); document.write( "a^2 +2a + 1/3 = 0
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\n" ); document.write( "a^2 +2a + 1 = (-1/3) + 1
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\n" ); document.write( "(a+1)^2 = 2/3
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\n" ); document.write( "a = -1 + square root(2/3) = -0.18
\n" ); document.write( "a = -1 - square root(2/3) = -1.82
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\n" ); document.write( "check these values
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\n" ); document.write( "f(-0.18) = 2.53
\n" ); document.write( "f(1.82) = 2.53
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\n" ); document.write( "interval is [-0.18, 1.82]
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\n" ); document.write( "note the interval [-1.82, 0.18] has the local max 4 at a=-1, therefore we exclude this interval
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