If y = x^1/x , x gt 0 , then displaystyledy/dx = 0 at

The correct answer is Option (4) → \(x = e\)

The variable is in both the base and the exponent, so we use logarithmic differentiation:

    \( \displaystyle \log_e y = \frac{\log_e x}{x} \)

Differentiating, with the quotient rule on the right:

    \( \displaystyle \frac{1}{y}\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \log_e x}{x^{2}} \)

    \( \displaystyle \frac{dy}{dx} = x^{1/x} \cdot \frac{1 - \log_e x}{x^{2}} \)

Since \(x^{1/x} \gt 0\) and \(x^{2} \gt 0\), the derivative is zero only when

    \( 1 - \log_e x = 0 \)
    \( x = e \)