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MHT CET202527 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

The Maximum value of ( 1 x )^x is

Options

  1. Ae ^ e
  2. Bx^5
  3. Cx
  4. D5-x^5

Correct answer

A. e ^ e

Step-by-step solution

Finding the maximum of f(x) = (1/x)^x Rewriting as f(x) = x^ -x and taking the natural logarithm gives y = -x x , where y = f(x) . Differentiate both sides with respect to x , yielding: 1 y dy dx = -( x + 1) via product rule on -x x dy dx = -y( x + 1) Substitute y = x^ -x to obtain the derivative: dy dx = -x^ -x ( x + 1) Setting dy dx = 0 for critical points, and noting x^ -x > 0 for x > 0 , solve: x + 1 = 0 x = -1 x = e⁻¹ The first derivative test confirms a local maximum at x = e⁻¹ : derivative is positive for x

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