Download App

Increasing and Decreasing Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions5 Marks
Question
Show that $\text{f}(\text{x})=\frac{1}{1+\text{x}^2}$ is decreases in the interval $[0,\infty)$ and increases in the interval $(-\infty,0].$
✓

Answer

Here, $\text{f}(\text{x})=\frac{1}{1+\text{x}^2}$Case 1:
Let $\text{x}_1,\text{x}_2\in (0,\infty)$ such that $\text{x}_1<\text{x}_2.$ Then,
$\text{x}_1<\text{x}_2$
$\Rightarrow\text{x}_1^2<\text{x}_2^2$ $\Rightarrow1+\text{x}_1^2<1+\text{x}_2^2$ $\Rightarrow\frac{1}{1+\text{x}_1^2}>\frac{1}{1+\text{x}_2^2}$ $\Rightarrow\text{f}(\text{x}_1)>\text{f}(\text{x}_2)\ \forall\ \text{x}_1,\text{x}_2\in(0,\infty)$ So, f(x) is decreasing on $(0,\infty).$Case 2:
Let $\text{x}_1,\text{x}_2\in (0,\infty]$ such that $\text{x}_1<\text{x}_2.$ Then, $\text{x}_1<\text{x}_2$ $\Rightarrow\text{x}_1^2>\text{x}_2^2$ $\Rightarrow1+\text{x}_1^2<1+\text{x}_2^2$ $\Rightarrow\frac{1}{1+\text{x}_1^2}<\frac{1}{1+\text{x}_2^2}$ $\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2)$ $\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2)\ \forall\ \text{x}_1,\text{x}_2\in(0,\infty]$ So, f(x) is increasing on $(0,\infty].$

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

Evaluate the following integrals:$\int^\limits{\text{a}}_{-\text{a}}\sqrt{\frac{\text{a}-\text{x}}{\text{a}+\text{x}}}\text{ dx}$
→
A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is $2\ m$ and volume is $8m^3.$ If building of tank cost $70$ per square metre for the base and Rs $45$ per square matre for sides, what is the cost of least expensive tank?
→
Evaluate the following integrals:$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin\text{x}|\sin\text{x}|\text{dx}$
→
Using differentials, find the approximate values of the following:
$\cos\Big(\frac{11\pi}{36}\Big)$
→
If $(\cos\text{x})^{\text{y}}=(\tan\text{y})^{\text{x}},$ Prove that $\frac{\text{dy}}{\text{dx}}=\frac{\log\tan\text{y}-\text{y}\tan\text{x}}{\log\cos\text{x}-\text{x}\sec\text{y cosec y}}$
→
Find one-parameter families of solution curves of the following differential equation: (or solve the following differential equation)$\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=\text{e}^{\sin\text{x}}\cos\text{x}$
→
Evaluate the following intregals:
$\int\frac{\cos\text{x}}{\cos3\text{x}}\ \text{dx}$
→
Show that the function $\text{f(x)}=\begin{cases}|2\text{x}-3||\text{x}|, & \text{x}\geq1\\\sin\Big(\frac{\pi\text{x}}{2}\Big),& \text{x}>1\end{cases}$ is continuous but not differentiable at x = 1.
→
Find the matrix A such that
$\begin{bmatrix}1&1\\0&1\end{bmatrix}\text{A}=\begin{bmatrix}3&3&5\\1&0&1\end{bmatrix}$
→
Find the angle between the following pairs of lines:$\frac{\text{x}-2}{3}=\frac{\text{y}+3}{-2},\text{z}=5$ and $\frac{\text{x}+1}{1}=\frac{2\text{y}-3}{3}=\frac{\text{z}-5}{2}$
→