find

Asked by jaindeepanshu000 | 23rd Jun, 2020, 10:06: PM

Expert Answer:

Given function is f(x) = x3 - 3(a - 2)x2 + 3ax + 7, for some a in R
It is given that f(x) is increasing in (0, 1] and decreasing in [1, 5)
To find the root of the equation {f(x)-14}/(x-1)2 = 0
As f(x) is increasing in the interval (0, 1] and decreasing in the interval [1, 5)
f'(x) >= 0 in (0, 1]
f'(x) <= 0 in [1, 5)
So, f'(x) = 0 at x = 1
f apostrophe left parenthesis x right parenthesis space equals space 3 x squared minus 6 left parenthesis a minus 2 right parenthesis x space plus space 3 a
f apostrophe left parenthesis 1 right parenthesis space equals space 0
rightwards double arrow space 3 left parenthesis 1 right parenthesis squared minus 6 left parenthesis a minus 2 right parenthesis left parenthesis 1 right parenthesis plus 3 a space equals space 0
rightwards double arrow space 3 minus 6 a plus 12 plus 3 a equals 0
rightwards double arrow space 15 minus 3 a equals 0
rightwards double arrow space a equals 5
f left parenthesis x right parenthesis equals x cubed minus 3 left parenthesis 5 minus 2 right parenthesis x squared plus 3 left parenthesis 5 right parenthesis x plus 7 equals x cubed minus 9 x squared plus 15 x plus 7
f left parenthesis x right parenthesis minus 14 equals x cubed minus 9 x squared plus 15 x minus 7 equals x cubed minus x squared minus 8 x squared plus 8 x plus 7 x minus 7
f left parenthesis x right parenthesis minus 14 equals x squared left parenthesis x minus 1 right parenthesis minus 8 x left parenthesis x minus 1 right parenthesis plus 7 left parenthesis x minus 1 right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis x squared minus 8 x plus 7 right parenthesis
f left parenthesis x right parenthesis minus 14 equals left parenthesis x minus 1 right parenthesis left parenthesis x squared minus 7 x minus x plus 7 right parenthesis equals left parenthesis x minus 1 right parenthesis left parenthesis x minus 1 right parenthesis left parenthesis x minus 7 right parenthesis
f left parenthesis x right parenthesis minus 14 equals left parenthesis x minus 1 right parenthesis squared left parenthesis x minus 7 right parenthesis
N o w comma space fraction numerator f left parenthesis x right parenthesis minus 14 over denominator left parenthesis x minus 1 right parenthesis squared end fraction equals 0
rightwards double arrow fraction numerator left parenthesis x minus 1 right parenthesis squared left parenthesis x minus 7 right parenthesis over denominator left parenthesis X minus 1 right parenthesis squared end fraction equals 0
rightwards double arrow x minus 7 equals 0
rightwards double arrow x equals 7

Answered by Renu Varma | 24th Jun, 2020, 12:26: PM

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