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+1 vote
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in Derivatives by (29.4k points)
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Write the maximum value of f(x) = \(\frac{log\,x}{x}\),if it exists.

1 Answer

+2 votes
by (28.3k points)
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Best answer

Let f(x) = \(\frac{log\,x}{x}\)

Clearly,

f(x) only defined for x > 0.

f'(x) = \(\frac{(1-log\,x)}{x^2}\) and

f"(x) = \(\frac{(2xlog\,x-3x)}{x^4}\) . (f"(x) < 0 for all x)

So,

f"(x) = 0

⇒ x= e and f''(e) < 0 

So, 

x = e is a point of maxima. 

Therefore,

f(e) = \(\frac{1}{e}\)

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