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Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

If f(x)=2+ _ -1 ^1 ( t x^2 2 + 9 x 14 ) f(t) d t , then:

Options

  1. ARolle's Theorem is applicable for y=f(x) in [-2,-1]
  2. B_ x 0 f(x)=0
  3. Cf is continuous and derivable on R
  4. Dmaximum value of f(x) does not exist

Correct answer

D. maximum value of f(x) does not exist

Step-by-step solution

f(x)=2+ x^2 2 _ -1 ^1 t f(t) d t+ 9 x 14 _ -1 ^1 f(t) d t Let A= 1 2 _ -1 ^1 t f(t) d t and B= 9 14 _ -1 ^1 f(t) d t aligned & f(x)=A x^2+B x+C & A= 1 2 _ -1 ^1 t (A t^2+B t+2 ) d t= ₀^1 B t^2 d t= B 3 3 A=B aligned and B= 9 14 _ -1 ^1 (A t^2+B t+2 ) d t= 9 14 2 ( A t^3 3 +2 t )₀^1 aligned & 3 A= 9 7 ( A 3 +2 )= 3 7 (A+6) & 21 A=3 A+18 A=1 and B=3 & f(x)=x^2+3 x+2 aligned Now, verify options.

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