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MHT CET202611 April 2026Evening ShiftMathematicsApplication of DerivativesActual

The value of k such that the function f(x) = x - x - kx + b is strictly decreasing for all real x is

Options

  1. Ak < 1
  2. Bk > 1
  3. Ck < 2
  4. Dk > 2

Correct answer

D. k > 2

Step-by-step solution

Given f(x) = x - x - kx + b Differentiating with respect to x , we get: f'(x) = x + x - k For f(x) to be strictly decreasing for all real x , we must have f'(x) 0 for all x R (with equality holding only at discrete points). x + x - k 0 k x + x We know that the maximum value of a x + b x is a^2 + b^2 . Maximum value of x + x = 1^2 + 1^2 = 2 Therefore, for the inequality to hold for all x , we must have: k 2 Among the given options, k > 2 satisfies this condition. Answer: k > 2

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