Sum

**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"`.

Concept: Maxima and Minima

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