Question
Prove that the function $\text{f}(\text{x})=\cos\text{x}$ is:
Strictly increasing in $(\pi,2\pi)$
Strictly increasing in $(\pi,2\pi)$
✓
Answer
$\text{f}(\text{x})=\cos\text{x}$
$\text{f}'(\text{x})=-\sin\text{x}$
Here,
$\pi<\text{x}<2\pi$
$\Rightarrow\sin\text{x}<0$ $[\because$ sine function is negative in third and fourth quadrent$]$
$\Rightarrow-\sin\text{x}>0$
$\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in(\pi,2\pi)$
So, f(x) is strictly decreasing on $(\pi,2\pi).$
$\text{f}'(\text{x})=-\sin\text{x}$
Here,
$\pi<\text{x}<2\pi$
$\Rightarrow\sin\text{x}<0$ $[\because$ sine function is negative in third and fourth quadrent$]$
$\Rightarrow-\sin\text{x}>0$
$\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in(\pi,2\pi)$
So, f(x) is strictly decreasing on $(\pi,2\pi).$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A point $P$ with position vector $\frac{-14 \hat{i}+39 \hat{j}+28 \hat{k}}{5}$ divides the line joining $A(-1,6,5)$ and $B$ in the
→ratio 3 : 2 then find the point B.
If in $\triangle A B C$ with usual notations $a=18, b=24, c=30$ then $\sin \frac{A}{2}=$
→Find the symbolic form of the following switching circuit. Construct its switching table and interpret it.
Evalute the following integrals:
$\int\frac{1-\sin2\text{x}}{\text{x}+\cos^2\text{x}}\text{dx}$
→$\int\frac{1-\sin2\text{x}}{\text{x}+\cos^2\text{x}}\text{dx}$
Construct a $2 × 2$ matrix $A = [a_{ij}]$ whose elements $a_{ij}$ are given by:
$\text{a}_\text{ij}=\frac{(\text{i}-2\text{j})^2}{2}$
→$\text{a}_\text{ij}=\frac{(\text{i}-2\text{j})^2}{2}$
Integrate the following functions w.r.t. x:
→$\frac{e^x}{x} \cdot\left[x(\log x)^2+2 \log x\right]$
A spherical soap bubble is expanding so that its radius is increasing at the rate of 0.02 cm/sec. At what rate is the surface area increasing, when its radius is 5 cm?
Find the area of the region lying in the first quadrant bounded by the curve $y^2=4 x, X$-axis and the lines $x=1$ and $x=4$.
→If $f(x)=2 x+5$ and $g(x)=x^2+1$ be two real functions, then describe the following functions:
fof
Also, show that fof $\neq f^2$
→fof
Also, show that fof $\neq f^2$
Evaluate the following : $\int\left(\frac{\cos x}{1-\cos x}\right) \cdot d x$
→