Download App

Increasing and Decreasing Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions5 Marks
Question
Show that $\text{f}(\text{x})=\cos\text{x}$ is a decreasing function on $(0,\pi),$ increasing in $(-\pi,0)$ and neither increasing nor decreasing in $(-\pi,\pi).$
✓

Answer

$\text{f}(\text{x})=\cos\text{x}$
Domain of $\cos\text{x}$ is $(-\pi,\pi).$
$\Rightarrow\text{f}'(\text{x})=-\sin\text{x}$
For $\text{x}\in(-\pi,0),\sin\text{x}<0$
$[\because$ Sine function is negative in third and fourth quadrant$]$
$\Rightarrow-\sin\text{x}>0$
$\Rightarrow\text{f}'(\text{x})>0$
So, $\cos\text{x}$ is increasing in $(-\pi,0).$
For $\text{x}\in(0,\pi),\sin\text{x}>0$
$[\because$ Sine function is positive in first and second quadrant$]$
$\Rightarrow-\sin\text{x}<0$
$\Rightarrow\text{f}'(\text{x})<0$
So, f(x) is decreasing on $(0,\pi).$
Thus, f(x) is neither increasing nor decreasing in $(-\pi,\pi).$

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 "Solve the Following Question.(5 Marks)" from Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the angle between the follwing pairs of lines:$\vec{\text{r}}=\big(4\hat{\text{i}}-\hat{\text{j}}\big)+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}\big)$ and $\vec{\text{r}}=\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}-\mu\big(2\hat{\text{i}}+4\hat{\text{j}}-4\hat{\text{k}}\big)$
→
$\frac{x-5}{4}=\frac{y-7}{-5}=\frac{z+3}{-5}$ and $\frac{x-8}{7}=\frac{y-7}{1}=\frac{z-5}{3}$
→
Find the co-factors of the elements of the following matrices: $\left[\begin{array}{ccc}1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1\end{array}\right]$
→
Find the angle between the follwing pairs of lines:$\vec{\text{r}}=\lambda\big(\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{j}}+\mu\big\{\big(\sqrt{3}-1\big)\hat{\text{i}}-\big(\sqrt{3}+1\big)\hat{\text{j}}+4\hat{\text{k}}\big\}$
→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_0\frac{\sin^{\text{n}}\text{x}}{\sin^\text{n}\text{x}+\cos^\text{n}\text{x}}\text{ dx},\text{ n}\in\text{N}$
→
Evaluate the following integrals:$\int^\limits{\frac{1}{2}}_{0}\frac{1}{(1+\text{x}^2)\sqrt{1-\text{x}^2}}\text{ dx}$
→
Consider $\text{f}:\text{R}_+\rightarrow[-5,\infty)$ given by $f(x) = 9x^2 + 6x - 5.$ Show that f is invertible with $\text{f}^{-1}(\text{x})=\frac{\sqrt{\text{x}+6}-1}{3}.$
→
Discuss the continuity of the function $\text{f(x)}=\begin{cases}2\text{x}-1,&\text{if }\text{ x}<2\\\frac{3\text{x}}{2},&\text{if }\text{ x}\geq2\end{cases}$
→
Differentiate the following functions with respect to x:
$(\text{x}^\text{x})\sqrt{\text{x}}$
→
Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{ x, y}(0)=1$
→