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COMEDK2024Evening ShiftMathematicsApplication of DerivativesActual

The turning point of the function y= a x-b (x-1)(x-4) at the point P(2,-1) is

Options

  1. Aa maximum
  2. Bboth maximum and a minimum
  3. Ca minimum
  4. Dneither a maximum nor a minimum

Correct answer

A. a maximum

Step-by-step solution

Given the function y = ax - b (x - 1)(x - 4) . Since the point P(2, -1) lies on the curve, we substitute x = 2 and y = -1 : -1 = 2a - b (2 - 1)(2 - 4) = 2a - b 1 (-2) = 2a - b -2 2 = 2a - b b = 2a - 2 The function becomes y = ax - (2a - 2) (x - 1)(x - 4) = a(x - 2) + 2 (x - 1)(x - 4) . Let u = x - 2 . Then x - 1 = u + 1 and x - 4 = u - 2 . y = au + 2 (u + 1)(u - 2) = au + 2 u^2 - u - 2 . For P(2, -1) to be a turning point, the derivative dy dx must be zero at x = 2 , which corresponds to u = 0 . Using the quotient

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