Algebra.Com's Answer #787657 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Consider the cubic function f(x) = ax^3 + bx^2 + cx + d. Determine the values of a, b, c, and d so that all of the following conditions are met.
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document.write( "a. f '(-1) = 1, f '(0) = -2
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document.write( "b. ๐โ๐๐๐ ๐๐ ๐ ๐๐๐๐๐ก ๐๐ ๐๐๐๐๐๐๐ก๐๐๐ ๐๐ก (1,0)
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document.write( "c. ๐โ๐ ๐ฆ ๐๐๐ก๐๐๐๐๐๐ก ๐๐ ๐กโ๐ ๐๐ข๐๐๐ก๐๐๐ ๐ฆ = ๐(๐ฅ) ๐๐ (0, โ3)
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document.write( "Find the constants ๐, ๐, ๐, d
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document.write( "The condition (c) gives us d = f(0) = -3.\r\n" );
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document.write( "So, the value of d is joust found.\r\n" );
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document.write( "The condition (a) f'(-1) = 1 leads to this equation\r\n" );
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document.write( " f'(-1) = 3a*(-1)^2 + 2b(-1) + c = 1, or\r\n" );
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document.write( " 3a - 2b + c = 1. (1)\r\n" );
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document.write( "The condition (a) f'(0) = -2 leads to this equation\r\n" );
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document.write( " f'(0) = 3a*0^2 + 2b*0 + c = -2, or\r\n" );
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document.write( " c = -2. (2) \r\n" );
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document.write( "So, the value of c is just found: c = -2.\r\n" );
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document.write( "The condition (b) means that the second derivative of f(x) has zero value at the point x= 1\r\n" );
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document.write( " f''(1) = 0, or\r\n" );
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document.write( " 6a + 2b = 0. (3)\r\n" );
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document.write( " By the way, the condition (b) also means that the point (1,0) lies on the curve y = f(x), or f(1) = 0.\r\n" );
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document.write( " It is a \"hidden\", an ADDITIONAL and an EXCESSIVE condition, which makes the solution impossible and non-existing, as you will see it later.\r\n" );
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document.write( "So, we just have two equation to determine two remaining unknown coefficients \"a\" and \"b\"\r\n" );
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document.write( " 3a - 2b = 3 (4) (obtained from (1) and (2))\r\n" );
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document.write( " 6a + 2b = 0 (5) (it is equation (3))\r\n" );
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document.write( "To solve this system, add equations (4) and (5). You will get then\r\n" );
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document.write( " 9a = 3,\r\n" );
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document.write( "which implies a =  .\r\n" );
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document.write( "Then from (4), 2b = -  = -2; hence, b = -1.\r\n" );
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document.write( "So, we just found all coefficients a = ; b = -1; c = -2 and d = -3. \r\n" );
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document.write( " y = f(x) =  .\r\n" );
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document.write( "CHECK\r\n" );
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document.write( " (a) f'(-1) = 3*(1/3)*(-1)^2 - 2*(-1) - 2 = 1 + 2 - 2 = 1. ! Correct !\r\n" );
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document.write( " (a) f'(0) = -2. ! Correct !\r\n" );
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document.write( " (b) f''(1) = 3*2*(1/3) - 2*1 = 2 - 2 = 0 ! Correct !\r\n" );
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document.write( " Now I should check the \"hidden\" condition that the point (1,0) lies on the curve.\r\n" );
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document.write( " f(1) = 3*(1/3) - 2*1 - 2*1 - 3 = 1 - 2 - 2 - 3 = -6 =/= 0 ! Incorrect !\r\n" );
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document.write( "The conclusion. As worded and presented, the given problem is self-contradictory and has no solution.\r\n" );
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document.write( "More explanations and post-solution note\r
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document.write( " Had the problem said \"there is a point of inflection at x = 1\", the problem would have a solution as shown above.\r\n" );
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document.write( " But in the form \"there is a point of inflection at (1,0)\", it brings a \"hidden\" condition that the point (1,0) lies\r\n" );
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document.write( " on the curve. With this hidden condition, we have 4+1 = 5 equations for four unknowns.\r\n" );
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document.write( " These 5 equations are INCONSISTENT, which makes the solution in this form IMPOSSIBLE and NON-EXISTING.\r\n" );
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document.write( "If after reading my post you will have questions, do not hesitate to post them to me.\r
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document.write( "If you do, then please refer to the ID number of this problem, which is 1163540.\r
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