If y = \(\rm\left(\frac{1}{x}\right)^x \), then value of \(\rm e^e\left(\frac{d^2 y}{d x^2}\right)_{x=e}\) is:

Calculation:

y = (1/x )^x

take log on both sides

log y = x log(1/x)

log y = x(log 1 - log x)

log y = x(-log x)

Differentiate with respect to y

1/y × dy/dx = -( 1 + log x)

dy/dx = -(1/x)^x(1 + log x)

again Differentiate with respect to x .

d²y/dx² = -(dy/dx(1 + logx) +y(1/x)

d2y/dx2 = -(-(1/x)^x(1+ log x)² + (1/x)^x+1)

\(\rm e^e\left(\frac{d^2 y}{d x^2}\right)_{x=e}\) = 4 -1/e

Hence, option 2 is correct.