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NIMCET 2012 Question Paper with solutions

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Question : 47 of 120

Marks: +1, -0

The function

x^x

decreases in the interval

(0 , e)
(0 , 1)
$\left(0, \frac{1}{e}\right)$
None of these

Solution:

Let y =

x^x

Taking log on both sides,

we get log y = x log x

On differentiating

\frac{1}{y} \frac{dy}{dx}

= x .

\frac{1}{x}

+ log x

\frac{dy}{dx}

= y (1 + log x) =

x^x

. (1 + log x) ... (i)

For decreasing of y,

Here ,

\frac{dy}{dx}

< 0

x^x

. (1 + log x) < 0 (but

x^x

≮ 0 and x > 0)

⇒ 1 + log x < 0

⇒ log x < - 1

⇒ log x < log

e^{-1}

⇒ x <

\frac{1}{e}

and x > 0

∴ x ∊

\left(0, \frac{1}{e}\right)

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