The maximum value of ≤ft(1/x)^{x}, x > 0 is

Correct Answer is : e1/ee¹ / e}e1/e

Correct Answer is : e1/ee¹ / e}e1/e

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MHTCET 15 Apr 2026 Shift 2 Paper

Question : 125 of 150

Marks: +1, -0

\left(\frac{1}{x}\right)^{x}, x > 0

Solution:

Concept:

To find the maximum of

y=\left(\frac{1}{x}\right)^{x}

for

x>0

, we use logarithms and differentiation.

Explanation:

Let

y=\left(\frac{1}{x}\right)^{x}

Take natural logarithm on both sides:

\ln y = x \ln\left(\frac{1}{x}\right)

Since

\ln\left(\frac{1}{x}\right)=-\ln x

, we get:

\ln y = -x \ln x

Because

\ln y

increases with

y

, maximising

y

is same as maximising

f(x)=-x\ln x

Differentiate

f(x)

f'(x) = -\left(\ln x + 1\right)

Set

f'(x)=0

-\left(\ln x + 1\right)=0 \Rightarrow \ln x = -1

So,

x = e^{-1} = \frac{1}{e}

Now substitute this value in the original expression:

\left(\frac{1}{x}\right)^{x} = \left(\frac{1}{1/e}\right)^{1/e} = e^{1/e}

Thus, the maximum value is

e^{1/e}

Answer:

e^{1/e}

, which is option C.

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