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Analyze the roots of the following equations: (i)2x^(3) - 9x^(2) + 12x - (9//2) = 0 (ii) 2x^(3) - 9x^(2) + 12x - 3 = 0 |
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Answer» Solution :Let `f(x) = 2X^(3) - 9x^(2) + 12x - (9//2)`. Then `f(x) = 6X^(2) - 18x +12` `= 6(x^(2)-3x+2) = 6(x- 1)(x-2)` Now `f'(x) = 0 rArr` x = 1 and x = 2 Also `f(1) = 2 - 9 + 12 - (9//2) gt 0` and`f(2) = 16 - 36 + 24 - (9//2) lt 0` HENCE, the graphs of the function `y= f(x)is as shown in the figure. (##CEN_ALG_C02_SLV_026_S01.png" width="80%"> As shown in the figure, the graph CUTS the x-axis at three distinct points Hence, equation`f(x) = 0` has three distinct roots. (ii) For `2x^(3) - 9x^(2) + 12x - 3 = 0,` `f(x) = 2x ^(3) - 9x + 12x - 3` `f'(x)= 0` `rArr6x^(2) - 18x + 12 = 0` or6 (x - 1) (x - 2) = 0 `rArr`x = 1and x = 2 Also ` f(1)= 2 - 9 + 12 - 3 = 2` and `f(2) = 16 - 36 + 24 - 3 = 1` Hence, the graphof y = f(x) is as shown in the figure. Thus , from the graph, we can see thatf(x) = 0 has only one real root, though y = f(x) has two turning points . |
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