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If function f x = a x + 7 ; 0 ≤ x < 2 b x + 1 ; x ≥ 2 is differentiable ∀ x ≥ 0 , then 2 a + 4 b is equal to

Options

  1. A1
  2. B5
  3. C4
  4. D9

Correct answer

A. 1

Step-by-step solution

f x must be continuous and differentiable at x = 2 For continuity at x = 2 f 2 - = f 2 + = f 2 a 2 + 7 = b 2 + 1 &#8658; 3 a = 2 b + 1 &#8230; i For differentiability at x = 2 f &#8242; x = a 2 x + 7 ; 0 &#8804; x &#60; 2 b ; x &#8805; 2 f &#8242; 2 &#8722; = f &#8242; 2 + a 2 2 + 7 = b &#8658; a = 6 b &#8230; i i From i &#160; &#38; &#160; ( i ) , we get, 18 b = 2 b + 1 &#8658; b = 1 16 2 a + 4 b = 12 b + 4 b = 16 b = 1

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