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Manipal MET2010MathematicsDifferentiation

If f(x)=(a x+b) x+(c x+d) x , then the values of a, b, c and d such that f^ (x)=x x for all x , are

Options

  1. Ab=c=0, a=d=1
  2. Bb=d=0, a=c=1
  3. Cc=d=0, a=b=1
  4. DNone of the above

Correct answer

A. b=c=0, a=d=1

Step-by-step solution

We have, f(x)=(a x+b) x+(c x+d) x aligned f^ (x)=a x+(a x+b) x & +c x -(c x+d) x aligned But f^ (x)=x x for all x (given) aligned & x x=(a-d) & x+(b+c) x+a x x-c x x aligned Equating the coefficients of x, x, x x and x x , we get aligned & a-d=0, b+c=0, a=1, c=0 & b=c=0 and a=d=1 aligned

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