1.

Analyze the roots of the following equations: (i)2x^(3) - 9x^(2) + 12x - (9//2) = 0 (ii) 2x^(3) - 9x^(2) + 12x - 3 = 0

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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