Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Sketch the curve, $y = \sqrt{x - [x]}$ (where $[\cdot]$ denotes the greatest integer function).

Accepted Answer

Given:

$y = \sqrt{x - [x]};$ where $[*]$ denotes the greatest integer function

We know that,

$[x] = \{\begin{cases} \begin{matrix} - 3; - 3 \leq x < - 2 \\ - 2; - 2 \leq x < - 1 \\ - 1; - 1 \leq x < 0 \\ 0; 0 \leq x < 1 \\ 1; 1 \leq x < 2 \\ 2; 2 \leq x < 3 \end{matrix} \\ 3; 3 \leq x < 4 \end{cases}$

and so on

So $f (x) = x - [x]$ becomes

$x - [x] = \{\begin{cases} \begin{matrix} x + 3; - 3 \leq x < - 2 \\ x + 2; - 2 \leq x < - 1 \\ x + 1; - 1 \leq x < 0 \\ x; 0 \leq x < 1 \\ x - 1; 1 \leq x < 2 \\ x - 2; 2 \leq x < 3 \end{matrix} \\ x - 3; 3 \leq x < 4 \end{cases}$

Again we know that

$x - [x] = \{x\}$

Where $[*]$ denotes greatest integer function and $\{*\}$ denotes the fractional part function

$\{x\} = \{\begin{cases} \begin{matrix} x + 3; - 3 \leq x < - 2 \\ x + 2; - 2 \leq x < - 1 \\ x + 1; - 1 \leq x < 0 \\ x; 0 \leq x < 1 \\ x - 1; 1 \leq x < 2 \\ x - 2; 2 \leq x < 3 \end{matrix} \\ x - 3; 3 \leq x < 4 \end{cases}$

As shown in the above figure, $y = \{x\}$ is a periodic function whose fundamental period is $1$ and domain: $x \in R$ and range; $y \in [0, 1)$

We know that for $y = \sqrt{x}$ , $x \geq 0$ and its graph is

Now we want to draw the graph of

$y = \sqrt{x - [x]}$

or $y = \sqrt{\{x\}}$

At every integer, the value of the function is zero.

As, $\{x\} \in [0, 1) \Rightarrow \sqrt{\{x\}} \in [0, 1)$

Hence, the minimum value of the function is $' 0'$ .

As we know $y = \{x\}$ is a periodic function then the graph of $y = \sqrt{x - [x]}$ will be

This is the required graph of $y = \sqrt{x - [x]}$ .

Sketch the curve, $y = \sqrt{x - [x]}$ (where $[\cdot]$ denotes the greatest integer function).

Important Questions on Curvature and Transformations

Learn and Prepare for any exam you want !

Sketch the curve, y=x-x (where · denotes the greatest integer function).

Important Questions on Curvature and Transformations

Learn and Prepare for any exam you want !

Sketch the curve, $y = \sqrt{x - [x]}$ (where $[\cdot]$ denotes the greatest integer function).