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Question
Determine the maximum and minimum value of the following function.
f(x) = x log x
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Solution
f(x) = x log x
∴ f'(x) =`"x" "d"/"dx" (log "x") + log "x" "d"/"dx" ("x")`
`= "x" xx 1/"x" + log "x" xx 1 = 1 + log "x"`
and f''(x) = `0 + 1/"x" = 1/"x"`
Consider, f'(x) = 0
∴ 1 + log x = 0
∴ log x = - 1
∴ log x = - log e = log e-1 = log `(1/"e")`
∴ x = `1/"e"`
For x = `1/"e"`
`f''(1/"e") = 1/(1/"e") = "e" > 0`
∴ f(x) attains minimum value at x = `1/"e"`.
∴ Minimum value = `"f"(1/"e") = 1/"e" log (1/"e") = 1/"e" log "e"^-1`
`= ((- 1)/"e") (1) = ((- 1)/"e")`
∴ The function f(x) has minimum value `(-1)/"e"` at x = `1/"e"`.
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