GATE AG 2008 • Q5 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Maxima and minima of function with several independent variables

Question Statement:

If loge(y) = −x loge(x), then the maximum value of y is

Options:

  • (A) e
  • (B) e^(x²)
  • (C) e^(e⁻¹)
  • (D) e⁻ˣ

Official Answer Key:

C

Step-by-Step Solution & Explanation

y = x^(−x) (x>0). ln y = −x ln x. Differentiate: (1/y)(dy/dx) = −(ln x + 1). Setting dy/dx=0 gives ln x = −1, x = 1/e. Then y_max = (1/e)^(−1/e) = e^(1/e). This is the maximum since d²y/dx² < 0 there.

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