Question

Write the minimum value of f(x) = x^x.

Answer 1

Let y = x^x. 

Take antilog on both sides 

ln y =x × ln x. 

Let us differentiate and find f'(x) = 0

⇒ \(\frac{1}{y}\times \frac{dy}{dx}\) = ln x + 1

⇒ \(\frac{dy}{dx}\) = y \(\times\)(ln x + 1)

⇒ f ’(x) = x^x × (ln x + 1)

f’(x) = 0

⇒ x = 0, x = \(\frac{1}{e}\)

But ln x is not defined at x = 0

Therefore, 

Minima occur at  x = \(\frac{1}{e}\).

So,

f(\(\frac{1}{e}\)) = (\(\frac{1}{e}\))^{\(\frac{1}{e}\)}