If f(x)=x^x , then f (x) is increasing in interval :

Correct Answer is : [0,1e]≤ft[0,1/e][0,e1]

Correct Answer is : [0,1e]≤ft[0,1/e][0,e1]

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BITSAT 2017 Paper

Section: Mathematics

Question : 98 of 150

Marks: +1, -0

f(x)=x^x

, then f (x) is increasing in interval :

Solution:

A function f(x) is said to be increasing function in [a, b] if

f'(x) > 0

in [a, b].

Given

f(x)=x^x

.... (i)

Differentiate equation (i)

f'(x)=x^{x}(1+\log x)

Put

f'(x)=0

0=x^{x}(1+\log x)

\Rightarrow x=0,\log x=-1\Rightarrow x=e^{-1}

\Rightarrow x=\frac{1}{e},0

Now in

\left[0,\;\;\frac{1}{e}\right],f^{I}(x)>0

∴ f(x) is increasing in interval

\left[0,\frac{1}{e}\right]

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