Download App

Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions3 Marks
Question
Without using the derivative, show that the function f(x) = |x| is
  1. Strictly increasing in $(0,\infty)$
  2. Strictly decreasing in $(-\infty,0)$
✓

Answer

We have, $\text{f}(\text{x})=|\text{x}|=\begin{cases}\text{x},&\text{x}>0\\\text{-x},&\text{x}<0\end{cases}$
  1. Let $\text{x}_1,\text{x}_2\in(0,\infty)$ and $\text{x}_1>\text{x}_2$
$\Rightarrow\text{f}(\text{x}_1)>\text{f}(\text{x}_2)$
So, f(x) is increasing in $(0,\infty).$
  1. Let $\text{x}_1,\text{x}_2\in (-\infty,0)$ and $\text{x}_1>\text{x}_2$
$\Rightarrow-\text{x}_1<-\text{x}_2$
$\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2)$
So, f(x) is decreasing on $(-\infty,0).$

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 "3 Marks Question" from Increasing and Decreasing Functions→Increasing and Decreasing Functions — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}1&2\\0&3 \end{bmatrix}$ is written as B + C, where B is a symmetric matrix and C is a skew- symmetric matrix, then B is equal to.
→
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector $2 \hat{i}-\hat{j}+4 \hat{k}$ and is in the direction $\hat{i}+2 \hat{j}-\hat{k}$
→
Find gof and fog when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by: $f(x) = 2x + 3$ and $g(x) = x^2 + 5$
→
Integrate the function: $\frac{{\left( {x + 1} \right){{\left( {x + \log x} \right)}^2}}}{x}$
→
A bill for Rs. 7650 was drawn on 8th March 2005 at 7 months. It was discounted on 18 May 2005 and the holder of the bill received Rs. 7497. What rate of interest did the banker charge?
→
Find the values of $p$ so that the lines $\frac{1-\text{x}}{3}=\frac{7\text{y}-14}{2\text{p}}=\frac{\text{z}-3}{2}\ \text{and}\ \frac{7-7\text{x}}{3\text{p}}=\frac{\text{y}-5}{1}=\frac{6-\text{z}}{5}$ are at right angles.
→
If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (-4, 3, -6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
→
The cartesian equation of a line AB are $\frac{2\text{x}-1}{\sqrt{3}}=\frac{\text{y}+2}{2}=\frac{\text{z}-3}{3}.$ Find the direction cosines of a line parallel to AB.
→
Find the inverse of $5$ under multiplication modulo $11$ on $Z_{11}.$
→
Prove that the Greatest Integer Function f: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
→