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JEE Main202226 Jul 2022Morning ShiftMathematicsApplication of DerivativesActual

Let f x = x 3 - x 2 + 10 x - 7 , x ≤ 1 - 2 x + log 2 b 2 - 4 , x > 1 Then the set of all values of b , for which f x has maximum value at x = 1 , is:

Options

  1. A- 6 , - 2
  2. B2 , 6
  3. C- 6 , - 2 ) ∪ ( 2 , 6
  4. D- 6 , - 2 ) ∪ ( 2 , 6

Correct answer

C. - 6 , - 2 ) ∪ ( 2 , 6

Step-by-step solution

Given, f x = x 3 - x 2 + 10 x - 7 , x ≤ 1 - 2 x + log 2 b 2 - 4 , x > 1 Now f 1 = 3 For x < 1 , f ' x = 3 x 2 - 2 x + 10 > 0 ⇒     f x is increasing For x > 1 , f ' x < 0 ⇒ function is decreasing. So, at x = 1 there is a point of maxima So, lim x → 1 + f x = - 2 + log 2 b 2 - 4 For maximum value at x = 1 ⇒ 3 ≥ - 2 + log 2   b 2 - 4 ⇒ 32 ≥ b 2 - 4 > 0 ⇒ b ∈ [ - 6 , - 2 ) ∪ ( 2 , 6 ]

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