The maximum value of frac{log x}{x}, 0

Question

(D) Let $y = \frac{\log x}{x}$.Find the derivative with respect to $x$ using the quotient rule:$\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \log x \cd

Vedclass · Accepted Answer

$e^{-1}$

Vedclass · Answer

(D) Let $y = \frac{\log x}{x}$.Find the derivative with respect to $x$ using the quotient rule:$\frac{dy}{dx} = \frac{x \cdot \frac{1}{x} - \log x \cdot 1}{x^2} = \frac{1 - \log x}{x^2}$.For critical points,set $\frac{dy}{dx} = 0$:$1 - \log x = 0 \Rightarrow \log x = 1 \Rightarrow x = e$.To verify the maximum,find the second derivative:$\frac{d^2y}{dx^2} = \frac{x^2(-\frac{1}{x}) - (1 - \log x)(2x)}{x^4} = \frac{-x - 2x + 2x \log x}{x^4} = \frac{2 \log x - 3}{x^3}$.At $x = e$,$\frac{d^2y}{dx^2} = \frac{2(1) - 3}{e^3} = -\frac{1}{e^3} The maximum value is $y(e) = \frac{\log e}{e} = \frac{1}{e} = e^{-1}$.

The maximum value of $\frac{\log x}{x}, 0 < x < \infty$ is

Similar Questions

$A$ manufacturer produces $x$ items per week at a total cost of Rs $(x^2+78x+2500)$. The price per unit is given by $8x = 600 - p$,where $p$ is the price of each unit. Then the maximum profit obtained is

$A$ box open from the top is made from a rectangular sheet of dimensions $a \times b$ by cutting squares of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum,then $x$ is equal to:

The function $f$ is defined by $f(x) = x^p (1 - x)^q$ for all $x \in R$,where $p, q$ are positive integers. The function has a maximum value for $x$ equal to:

The maximum value of $xy$ subject to $x+y=7$ is

What is the minimum value of the sum of a real number $x$ and its reciprocal?

Vedclass Test Series

Exam Paper Generator

Online Exam Module