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The set of all the possible values of ‘a’ for which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum values at some x1

The set of all the possible values of ‘a’ for which the function f(x)=5+(a-2)x+(a-1)x^2-x^3 has a local minimum values at some x1

Grade:12th pass

1 Answers

Vikas TU
14149 Points
3 years ago
Dear Student,
f(x) = 5 + (a-2)x + (a-1)x2 – x3.
The extremum occurs when f'(x) = (a-2) + 2(a-1)x – 3x2 = 0.
For x1, the condition for extremum is 3x12 -  2(a-1)x1 – (a-2) = 0.
For this x1 to be minimum, f''(x1) gretare than equals   0.
6x1 – 2(a-1) ≥  0 => 6x1 ≥ 2(a-1) => 3x1 ≥ a-1 =>  3x1 + 1 ≥ a.
Henc ethe possible values of a would be in term of x =>
a less than equalls to all the values 3x + 1.
Cheers!!
Regards,
Vikas (B. Tech. 4th year
Thapar University)

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