The maximum value of logx/x is

D. 1/e. Step 1: Let f(x)=logx/x for x>0. Then f′(x)=1-logxx^2. Step 2: f′(x)=0 gives logx=1, so x=e. Also f′>0 for x e, so x=e is a maximum. Step 3: f(e)=loge/e=1/e. Check: f(1)=0 and f(e^2)=2e^2, which is smaller than 1/e (see figure). Trap: Option (B) is 2/e, twice the true maximum; the graph never rises that high.

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AMU MCA 2016

The maximum value of $\frac{l o g x}{x}$ is

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — $\frac{1}{e}$

Detailed Solution

Let $f (x) = \frac{l o g x}{x}$ for $x > 0$ . Then $f' (x) = \frac{1 - l o g x}{x ^{2}}$ .

$f' (x) = 0$ gives $l o g x = 1$ , so $x = e$ . Also $f' > 0$ for $x < e$ and $f' < 0$ for $x > e$ , so $x = e$ is a maximum.

$f (e) = \frac{l o g e}{e} = \frac{1}{e}$ .

✓ Check

f (1) = 0

and

f (e^{2}) = \frac{2}{e ^{2}}

, which is smaller than

\frac{1}{e}

(see figure).

⚠ Common trap

Option (B) is

\frac{2}{e}

, twice the true maximum; the graph never rises that high.

Quick tip

$\frac{d}{d x} (\frac{l o g x}{x}) = 0 \Rightarrow x = e, maximum value = \frac{1}{e}$

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Parabola — Preparation

This question tests your understanding of Parabola, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix), so its eccentricity is exactly 1. The standard form y^2 = 4ax has its vertex at the origin, focus at (a, 0), directrix x = -a, and latus rectum of length 4a. The other standard forms open left, up or down. Exam questions test identifying the focus, directrix, axis and latus rectum, the parametric point (at^2, 2at), focal distances, and equations of tangents and normals. A key result is that the line y = mx + c touches y^2 = 4ax when c = a/m. Drawing a quick sketch of which way the parabola opens prevents most sign errors. Practise the parabola questions below, each with a step-by-step solution.

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