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Question
The maximum value of `(1/x)^x` is ______.
Options
e
ex
`"e"^(1/"e")`
`(1/"e")^(1/"e")`
ee
Solution
The maximum value of `(1/x)^x` is `underlinebb(e^(1/e))`.
Explanation:
Let f(x) = `(1/x)^x`
Taking log on both sides, we get
log [f (x)] = `x log 1/x`
⇒ log [f (x)] = `x log x^-1`
⇒ 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) [1 + log x]`
⇒ f'(x) = `- (1/x)^x [1 + log x]`
For local maxima and local minima f'(x) = 0
`-(1/x)^x [1 + log x]` = 0
⇒ `(1/x)^x [1 + log x]`= 0
`(1/x)^x ≠ 0`
∴ 1 + log x = 0
⇒ log x = – 1
⇒ x = e–1
So, x = `1/"e"` is the stationary point.
Now f'(x) = `-(1/x)^x [1 + log x]`
f"(x) = `-[(1/x)^x (1/x) + (1 + log x) * "d"/"dx" (x)^x]`
f"(x) = `-[("e")^(1/"e") ("e") + (1 + log 1/"e") "d"/"dx" (1/"e")^(1/"e")]`
x = `1/"e"`
= `-"e"^(1/"e") 1 < 0` maxima
∴ Maximum value of the function at x = `1/"e"` is
`"f"(1/"e") = (1/(1/"e"))^(1/"e") = "e"^(1/"e")`
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