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COMEDK2024MathematicsApplication of DerivativesActual

If f(x)= a x+b x c x+d x is decreasing for all x , then

Options

  1. Aa b-c d>0
  2. Ba d-b c<0
  3. Ca d-b c>0
  4. Da b-c d<0

Correct answer

B. a d-b c<0

Step-by-step solution

Given f(x) = a x + b x c x + d x . To determine the condition for f(x) to be decreasing, we find the derivative f'(x) using the quotient rule ( u v )' = u'v - uv' v^2 . Let u = a x + b x and v = c x + d x . u' = a x - b x and v' = c x - d x . f'(x) = (a x - b x)(c x + d x) - (a x + b x)(c x - d x) (c x + d x)^2 . Expanding the numerator: Numerator = (ac x x + ad ^2 x - bc ^2 x - bd x x) - (ac x x - ad ^2 x + bc ^2 x - bd x x) . Numerator = ad ^2 x - bc ^2 x + ad ^2 x - bc ^2 x . Numerator = ad( ^2 x + ^2 x) - bc( ^

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