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Most Important Selected Qs for JEE AdvancedMathematicsDifferentiation

Consider a cubic, f(x)=a x^3+b x^2+c x+4, a, b, c R and f^ ( -2 3 )=0, f^ (0)=3 , f^ ( -2 3 )= 5 3 , and g=f⁻¹ , then:

Options

  1. Aa+b=c
  2. Ba b c=6
  3. C.(g(x) f(g(x)))^ |_ x=4 = 4 3
  4. D.(g(x) g(f(x)))^ |_ x=4 = 3 4

Correct answer

C. .(g(x) f(g(x)))^ |_ x=4 = 4 3

Step-by-step solution

aligned & f^ (x)=3 a x^2+2 b x+c f^ (0)=c=3 & f^ (x)=6 a x+2 b f^ ( -2 3 )=-4 a+2 b=0 b=2 a aligned f^ ( -2 3 )=3 a 4 9 +4 a ( -2 3 )+3= 5 3 aligned & 4 a 3 = 4 3 a=1, b=2 & f(x)=x^3+2 x^2+3 x+4 aligned d d x (g(x) f(g(x)))= d d x (x g(x))=x g^ (x)+g(x) aligned & f(0)=4 g(4)=0 & and g^ (f(x))= 1 f^ (x) g^ (4)= 1 f^ (0) = 1 3 aligned . (g(x) f(g(x)))^ |_ x=4 = 4 3

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