The set of real values of x for which $ f(x) = \frac {x}{log\, x}$ increasing, is

$f(x)= \frac{x}{\log x}$ $f'(x)=\frac{\log x \cdot 1-x \cdot \frac{1}{x}}{(\log x)^{2}}=\frac{(\log x-1)}{(\log x)^{2}}$ We know that, $f(x)$ is increasing (strictly) When $f^{\prime}(x)>0$ $\Rightarrow \frac{(\log x-1)}{(\log x)^{2}} >0 $ $\Rightarrow (\log x-1)>0$ $\Rightarrow \log x>1$ $\Rightarrow \log _{e} x>\log _{e} e$ $\Rightarrow x >e$ Hence, $x: x \geq e$

The set of real values of x for which f(x)=x?/logx increasing is

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Concepts Used:

Application of Derivatives

Various Applications of Derivatives-

Rate of Change of Quantities:

If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by

$\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}$

This is also known to be as the Average Rate of Change.

Increasing and Decreasing Function:

Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).

If for any two points x₁ and x₂ in the interval x such a manner that x₁ < x₂, there holds an inequality f(x₁) ≤ f(x₂); then the function f(x) is known as increasing in this interval.
Likewise, if for any two points x₁ and x₂ in the interval x such a manner that x₁ < x₂, there holds an inequality f(x₁) ≥ f(x₂); then the function f(x) is known as decreasing in this interval.
The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x₁) < f(x₂) for strictly increasing and f(x₁) > f(x₂) for strictly decreasing.

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