Find the maximum value of (1/x)*x …

For every real number (or) valued function f(x), the values of x which satisfies the equation f¹(x)=0are the point of it's local and global maxima or minima. This occus due to the fact that, at the point of maxima or minima, the curve of the function has a ,zero slope. We have function f(x)=(1​/x)^x We will be using the equation, y=(1/x)^x Taking in both sides we get ln y=−xlnx Differentiating both sides with respect to x. y.dy​/dx = −lnx − 1 dy​/dx=−y (ln x + 1) Equating dy​/dx  to 0, we get −y(ln x + 1)=0 Since y is an exponential function it can never be equal to zero, hence ln x + 1 = 0 ln x = −1 x=e^−1 So, for the maximum value we put x=e^−1 in f(x) to get the value of f(x) at the point. f(e^−1) = e^1/e Hence the maximum value of the function is e^1/e. ∴ So, the answer is option →B. {e^1/e}

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