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MHT CET202521 Apr 2025Morning ShiftMathematicsApplication of DerivativesActual

The function x^5-5 x^4+5 x^3-10 has a maximum, when x is equal to

Options

  1. A0
  2. B1
  3. C2
  4. D3

Correct answer

B. 1

Step-by-step solution

Critical points of f(x) = x^5 - 5x^4 + 5x^3 - 10 are found where the first derivative vanishes. The derivative is f'(x) = 5x^4 - 20x^3 + 15x^2 , which factors as 5x^2(x-1)(x-3) . Setting f'(x) = 0 yields critical values at x = 0 , x = 1 , and x = 3 . The second derivative is f''(x) = 20x^3 - 60x^2 + 30x . At x = 1 , f''(1) = -10 At x = 3 , f''(3) = 90 > 0 , indicating a local minimum. At x = 0 , f'(x) does not change sign, identifying it as an inflection point. The maximum occurs at x = 1 .

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