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KCET2010MathematicsDifferentiation

If the three function f(x), g(x) and h(x) are such that h(x)=f(x) g(x) and f^ (x) g^ (x)=c where c is constant, then f^ (x) f(x) + g^ (x) g(x) + 2 c f(x) g(x) is equal to

Options

  1. Ah^ (x) h^ (x)
  2. Bh ( x ) h ^ ( x )
  3. Ch^ (x) h(x)
  4. Dh(x) h^ (x)

Correct answer

C. h^ (x) h(x)

Step-by-step solution

Given, h(x)=f(x) g(x) and f^ (x) g^ (x)=c Now, h^ (x)=f^ (x) g(x)+f(x) g^ (x)h^ (x)=f^ (x) g(x)+f^ (x) g^ (x)+f^ (x) g^ (x)+f(x) g^ (x)h^ (x)=f^ (x) g(x)+f(x) g^ (x)+2 f^ (x) g^ (x)h^ (x)=f^ (x) g(x)+f(x) g^ (x)+2 c (i) Now, we find f^ (x) f(x) + g^ (x) g(x) + 2 c f(x) g(x) aligned &= f^ (x) g(x)+g^ (x) f(x)+2 c f(x) g(x) &= h^ (x) h(x) [ from Eq. (i)] aligned

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