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AP EAMCET202318 May 2023Evening ShiftMathematicsContinuity and DifferentiabilityActual

If f(x)= array ll a x^2-b x+2, & x < 3 b x^2-3, & x 3 array . is differentiable at every x R , then the area (in sq units) of the triangle formed by the line x a + y b =1 with the coordinate axes is

Options

  1. A175 81
  2. B175 27
  3. C35 27
  4. D125 27

Correct answer

B. 175 27

Step-by-step solution

f ( x ) is differentiable at every x R . f ( x ) will be differentiable at x =3 also. So, L.H.D = R.H.D.....(i) aligned & L.H.D = _ x 3 (2 a x-b)=6 a-b & R.H.D = _ x 3 (2 b x)=6 b & 6 a-b=6 b 6 a=7 b .....(ii) & f(x) is continuous on x=3 & L.H.L = R.H.L & 9 a-3 b+2=9 b-3 & 9 a-12 b=-5.....(iii) aligned Solving eq ^ n (ii) & (iii), we get : aligned & a= 35 9 , b= 10 3 & Area = 1 2 a b= 1 2 35 9 10 3 = 175 27 aligned

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