The maximum value of \(\rm \left(\dfrac{1}{x}\right)^x\)

Concept:

Steps to find the maxima and minima

1.find \(\rm \dfrac {dy}{dx}\) and equate to zero

2. find the value of x i. e. critical point

3. for the maximum value we put x in f(x) to get the value of f(x)  at the point.

Calculations:

Given, f(x) = \(\rm \left(\dfrac{1}{x}\right)^x\)

Consider, y = \(\rm \left(\dfrac{1}{x}\right)^x\)

Taking log on both side, we get

ln y = x ln(\(\rm \dfrac 1 x\))

⇒ln y = -x ln x

⇒(1/y) × \(\rm \dfrac {dy}{dx}= -ln\; x -1\)

Equating \(\rm \dfrac {dy}{dx} = 0\)

⇒\(\rm -ln\; x -1 = 0\)

⇒ x = \(\dfrac 1 e\)

So, for the maximum value we put x = \(\dfrac 1 e\) in f(x) to get the value of f(x)  at the point.

f(x) = e1/e

Hence, The maximum value of \(\rm \left(\dfrac{1}{x}\right)^x\) is e1/e