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

Evaluate the following functions : $\int \frac{1}{x+\sqrt{x}} \cdot d x$

Solve the following differential equations.

$\frac{d y}{d x}=- k$, where $k$ is a constant.

Find the separate equations of lines represented by the following equations : $2 x^2+2 x y-y^2=0$

Determine the order and degree of the following differential equations:

$\frac{d^4 y}{d x^4}+\sin \left(\frac{d y}{d x}\right)=0$

$\int_0^{\pi / 2} x \cdot \sin x \cdot d x$

Which of the following sentences are statements in logic? Justify. Write down the truth value of the statements :Please get me a glass of water.

The displacement of a particle at time t is given by $s = 2t^3 – 5t^2 + 4t – 3$. Find the velocity when ? $= 2\ sec$

Write the differential equation obtained emliminating the arbitrary constant $C$ in the equation $xy = C^2$.

The non zero vectors $\overline{ a }$ and $\overline{ b }$ are not collinear find the value of $\lambda$ and $\mu$ : if $\overline{ a }+3 \overline{ b }=2 \lambda \overline{ a }-\mu \overline{ b }$

Find the area of the region bounded by the following curves, X-axis, and the given lines:

$y=\sin x, x=0, x=\frac{\pi}{2}$