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Question
f(x) = xx has a stationary point at ______.
Options
x = e
x = `1/"e"`
x = 1
x = `sqrt("e")`
Solution
f(x) = xx has a stationary point at x = `1/"e"`.
Explanation:
We have f(x) = xx
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] = xx[1 + log x]
To find stationary point, f'(x) = 0
∴ xx[1 + log x] = 0
xx ≠ 0 ∴ 1 + log x = 0
⇒ log x = – 1
⇒ x = e–1
⇒ x = `1/"e"`
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