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MHT CET20242 May 2024Morning ShiftMathematicsApplication of DerivativesActual

If Rolle's theorem holds for the function f (x)=x^3+ b x^2+ ax +5 on [1,3] with c =2+ 1 3 , then the values of a and b respectively are

Options

  1. A-11,6
  2. B11,6
  3. C-11,-6
  4. D11,-6

Correct answer

D. 11,-6

Step-by-step solution

Since f (x) satisfies the Rolle's theorem, array ll & f (1)= f (3) & 1+ b + a +5=27+9 ~b +3 a +5 & 2 a +8 ~b =-26 & a +4 ~b =-13...(i) array array ll & f (x)=x^3+ b x^2+ a x+5 & f ^ (x)=3 x^2+2 ~b x+ a array Now, f^ (c)=0 aligned & f ^ (2+ 1 3 )=0 & 3 (2+ 1 3 )^2+2 ~b (2+ 1 3 )+ a =0 & 3 (4+ 4 3 + 1 3 )+4 ~b + 2 ~b 3 + a =0 & a +4 ~b + 2 ~b +12 3 +13=0 & -13+ 2 ~b +12 3 +13=0 [From (i)] & 2 ~b +12 3 =0 & ~b =-6 aligned Substituting b=-6 in (i), we get a=11

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