What is the minimum value of the function y=e^ln(x^x) ?

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We have the function y = e^ln(x^x)

take the log of both the sides

ln y = ln (e^ln(x^x))

=> ln y = ln(x^x)

=> ln y = x*ln x

differentiate both the sides

(1/y)dy/dx = ln x + x/x

dy/dx = y*(ln x + 1)

Equating dy/dx = 0

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We have the function y = e^ln(x^x)

take the log of both the sides

ln y = ln (e^ln(x^x))

=> ln y = ln(x^x)

=> ln y = x*ln x

differentiate both the sides

(1/y)dy/dx = ln x + x/x

dy/dx = y*(ln x + 1)

Equating dy/dx = 0

=> e^ln(x^x)*(ln x + 1) = 0

=> ln x + 1 = 0

=> ln x = -1

=> x = e^-1

For x = e^-1

f(x) = e^ln((1/e)^(1/e))

=> (1/e)^(1/e)

=> e^(-1/e)

Therefore the required minimum value of the function is e^(-1/e)

Approved by eNotes Editorial Team