For f(x)=e-/(2), where =(x-)2/(22) and σ, μ are constants, the maximum occurs at

GATE 2020 AE – Question 12

Engineering Mathematics · Calculus: Limits, continuity, differentiability, chain rule, maxima and minima, integration · 1 mark · Multiple choice

For $f(x)=e^{-\lambda}/(\sigma\sqrt{2\pi})$, where $\lambda=(x-\mu)^2/(2\sigma^2)$ and σ, μ are constants, the maximum occurs at

$x=\sigma$
$x=\sigma\sqrt{2\pi}$
$x=2\sigma^2$
$x=\mu$

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Show answer and explanation

Correct answer: (D) $x=\mu$

Explanation

For the intended Gaussian density σ > 0. The prefactor is a positive constant and $e^{-\lambda}$ decreases as λ increases. The squared deviation is minimized at x = μ, where λ = 0. Hence f is maximized at x = μ, D.

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