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JEE MainMathematicsApplication of Derivatives

Let f(x) = ₀^ x^2 - 6x t^2 - 7t + 12 2 + ^2 t dt . If m and n respectively are the number of points of local maxima and local minima of the function f(x) , then the ordered pair (m, n) is equal to

Options

  1. A(2, 3)
  2. B(3, 2)
  3. C(2, 2)
  4. D(3, 3)

Correct answer

A. (2, 3)

Step-by-step solution

Given f(x) = ₀^ x^2 - 6x t^2 - 7t + 12 2 + ^2 t dt Differentiating f(x) with respect to x using the Leibniz rule: f'(x) = (x^2-6x)^2 - 7(x^2-6x) + 12 2 + ^2(x^2-6x) d dx (x^2-6x) f'(x) = (x^2-6x)^2 - 7(x^2-6x) + 12 2 + ^2(x^2-6x) (2x-6) To find the critical points, we set f'(x) = 0 . The denominator 2 + ^2(x^2-6x) is always strictly positive. Thus, f'(x) = 0 (x^2-6x)^2 - 7(x^2-6x) + 12 = 0 or 2x-6 = 0 . From 2x-6 = 0 , we get x = 3 . For the other factor, let u = x^2-6x . The equation becomes: u^2 - 7u + 12 = 0 (u-

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