If $ f(x)\,=kx\,-\,\cos \,x $ is monotonically increasing for all $ x\in R, $ then

Given, $ f(x)=kx-\cos x $ and $ x\in R $ $ f'(x)=k+\sin x $ Since, the function $ f(x) $ is monotonically increase for all $ x\in R $ . $ \therefore $ $ f'(x)>0,\,\,\,\forall \,\,\,\,x\in R $ $ \Rightarrow $ $ f'\left( \frac{\pi }{2} \right)>0 $ $ \Rightarrow $ $ k+\sin \frac{\pi }{2}>0 $ $ \Rightarrow $ $ k+1>0 $ $ \Rightarrow $ $ k>-1 $

If $ f(x)\,=kx\,-\,\cos \,x $ is monotonically inc

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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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If $ f(x)\,=kx\,-\,\cos \,x $ is monotonically increasing for all $ x\in R, $ then

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