The function xx decreases on the interva

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Applications of Derivatives

Question:

The function x^x decreases on the interval

Options:

(0, e)

(0, 1)

$(0, \frac{1}{e})$

None of these

Correct Answer:

$(0, \frac{1}{e})$

Explanation:

$f(x)=x^x ⇒ \log f(x)=x \log x$

⇒ $f'(x) = x^2[1+\log x]=x^x(\log e+\log x)=x^x\left(\log e^x\right)$

∴ for 0 < x < $\frac{1}{e}$ ⇒ ex < 1 ⇒ log e x < 0

⇒ f'(x) < 0 ⇒ f(x) is decreasing on $\left(0, \frac{1}{e}\right)$.