Show that f(x) = x – x is increasing for all x. - Vidyadip

Question

Show that $f(x) = x –\cos x$ is increasing for all $x.$

✓

Answer

$ f(x)=x-\cos x$
$\therefore f^{\prime}(x)=1+\sin x $
Note that $-1 \leq \sin x \leq 1, \forall x$
$ \therefore-1+1 \leq 1+\sin x \leq 1+1, \forall x$
$\therefore 0 \leq 1+\sin x \leq 2, \forall x $
i.e., $f^{\prime}(x) \geq 0$ for all $x$.
Hence, $f(x)$ is increasing for all $x$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Answer the following questions in short." from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Diffrentiate the following w. r. t. x.

$\cot ^{-1}\left[\cot \left(e^{ x 2}\right)\right]$

Evaluate: $\int_0^1(x+1)^2 d x$

Integrate the following functions w.r.t. x:

$\left[2+\cot x-\operatorname{cosec}^2 x\right] e^x$

Determine the truth values ofp and q in the following cases : (p ∧ q) is F and (p ∧ q) → q is T

Let X represent the difference between a number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$(\text{y"})^2+(\text{y})^3+\sin\text{y}=0$

Given an example of:
A triangular matrix.

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\Big(\frac{\text{dy}}{\text{dx}}\Big)^3-4\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+7\text{y}=\sin\text{x}$

Verify that the combined equation of lines 2x + 3y = 0 and x – 2y = 0 is a homogeneous equation
of degree two.

Find the Cartesian equations of the line passing through the point $\mathrm{A}(2,1,-3)$ and perpendicular
to vectors $\bar{b}=\hat{i}+\hat{j}+\hat{k}$ and $\bar{c}=\hat{i}+2 \hat{j}-\hat{k}$