Question

f(x) = x^x has a stationary point at ______.

Options

• x = e

• x = `1/"e"`

• x = 1

• x = `sqrt("e")`

Answer 1

f(x) = x^x has a stationary point at x = `1/"e"`.

Explanation:

We have f(x) = x^x 

Taking log of both sides, we have

log f(x) = x log x

Differentiating both sides w.r.t. x, we get

`1/("f"(x)) * "f'"(x) = x * 1/x + log x * 1`

⇒ f'(x) = f(x)[1 + log x] = x^x[1 + log x]

To find stationary point, f'(x) = 0

∴ x^x[1 + log x] = 0

x^x ≠ 0 ∴ 1 + log x = 0

⇒ log x = – 1

⇒ x = e^–1

⇒ x = `1/"e"`