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Question
Find the local extrema for the following functions using second derivative test:
f(x) = x log x
Solution
f'(x) = `x - 1/4 + log = 1 + log x`
For maximum or minimum
f'(x) = 0
⇒ 1 + log x = 0
⇒ log x = –1
x = `"e"^-1 = 1/"e"`
f”(x) = `1/x`
At x = `1/"e", f”(x) > 0
⇒ f(x) attains minimum.
∴ Local minimum `"f"(1/"e")`
= `1/"e" log (1/"e")`
= `1/"e"(- 1)`
= `- 1/"e"`
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