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JEE MainMathematicsApplication of Derivatives

Let f(x) = 4 x-2 + 3 7-x be a real valued function. The number of integers in the range of f(x) is

Options

  1. A5
  2. B2
  3. C9
  4. D4

Correct answer

A. 5

Step-by-step solution

The domain of the function is determined by x-2 0 and 7-x 0 , which gives x [2, 7] . We find the maximum value of f(x) using the Cauchy-Schwarz inequality: (4 x-2 + 3 7-x )^2 (4^2 + 3^2)(( x-2 )^2 + ( 7-x )^2) (f(x))^2 (16 + 9)(x-2 + 7-x) (f(x))^2 25(5) = 125 Thus, the maximum value of f(x) is 125 . Since f(x) is the sum of two strictly concave functions, its minimum value on the closed interval [2, 7] must occur at one of the endpoints. Evaluating at the endpoints: f(2) = 4(0) + 3 5 = 45 f(7) = 4 5 + 3(0) = 80 The

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