The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[ x^2 - x - \frac{1}{2} \right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is _________

Explore Exams

All Exams Test series for 1 year @ ₹349 only

Question

The number of points in the interval $[2, 4]$, at which the function $f(x) = \left[ x^2 - x - \frac{1}{2} \right]$, where $[\cdot]$ denotes the greatest integer function, is discontinuous, is __________.

Analyzing Discontinuity Points for Greatest Integer Functions

The greatest integer function, denoted by $ [\cdot] $, results in a discontinuity whenever its argument evaluates to an integer.

Identifying Discontinuity Conditions

For the function $ f(x) = \left[ x^2 - x - \frac{1}{2} \right] $, let $ g(x) = x^2 - x - \frac{1}{2} $. Discontinuities occur when $ g(x) $ is an integer.

Determining the Range of $ g(x) $ in $ [2, 4] $

To find points of discontinuity within the interval $ [2, 4] $, we first determine the range of $ g(x) $ over this interval. As $ g(x) $ represents an upward-opening parabola with its vertex at $ x = 1/2 $, it is strictly increasing on $ [2, 4] $.

Minimum value: $ g(2) = 2^2 - 2 - \frac{1}{2} = 4 - 2 - 0.5 = 1.5 $.
Maximum value: $ g(4) = 4^2 - 4 - \frac{1}{2} = 16 - 4 - 0.5 = 11.5 $.
The range of $ g(x) $ for $ x \in [2, 4] $ is $ [1.5, 11.5] $.

Finding Integer Values in the Range

Discontinuities happen when $ g(x) $ equals an integer in $ [1.5, 11.5] $. The integers are:

$ \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} $

Solving $ g(x) = k $ for $ x $

We solve the equation $ x^2 - x - \frac{1}{2} = k $ for each integer $ k $ identified above. Rearranging gives:

$ x^2 - x - \left(\frac{1}{2} + k\right) = 0 $

Applying the quadratic formula $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $:

$ x = \frac{1 \pm \sqrt{(-1)^2 - 4(1)\left(-\frac{1}{2} - k\right)}}{2} = \frac{1 \pm \sqrt{1 + 2 + 4k}}{2} $ $ x = \frac{1 \pm \sqrt{3 + 4k}}{2} $

Checking Solutions within Interval $ [2, 4] $

We consider the positive root $ x = \frac{1 + \sqrt{3 + 4k}}{2} $.

For $ k=2 $, $ x = \frac{1 + \sqrt{11}}{2} \approx 2.158 $, which is in $ [2, 4] $.
For $ k=11 $, $ x = \frac{1 + \sqrt{47}}{2} \approx 3.928 $, which is in $ [2, 4] $.

All integers $ k $ from 2 to 11 yield a distinct solution for $ x $ within $ [2, 4] $ because $ g(x) $ is increasing. The negative root $ x = \frac{1 - \sqrt{3 + 4k}}{2} $ is always outside $ [2, 4] $ for $ k \ge 2 $.

Counting the Discontinuities

There are $ 11 - 2 + 1 = 10 $ integer values for $ g(x) $ in the range $ [1.5, 11.5] $. Each integer corresponds to a unique point of discontinuity for $ f(x) $ within the interval $ [2, 4] $.

Therefore, there are 10 points of discontinuity.

Was this answer helpful?

Important Questions from Differential Calculus

Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x - 2) + \alpha x^2 + 4x - \alpha$, $x > 2$, is ______
JEE Main
Mathematics
View Answer
Consider the following three statements for the function $f : (0, \infty) \rightarrow \mathbb{R}$ defined by $f(x) = |\log_e x| - |x - 1|$:
(I) $f$ is differentiable at all $x > 0$.
(II) $f$ is increasing in $(0, 1)$.
(III) $f$ is decreasing in $(1, \infty)$.
Then.
JEE Main
Mathematics
View Answer
If the domain of the function $f(x) = \sin^{-1} \left( \frac{1}{x^2 - 2x - 2} \right)$ is $(-\infty, \alpha] \cup [\beta, \gamma] \cup [\delta, \infty)$, then $\alpha + \beta + \gamma + \delta$ is equal to
JEE Main
Mathematics
View Answer
Let $[t]$ denote the greatest integer less than or equal to $t$. If the function
$f(x) = \begin{cases} b^2 \sin \left( \frac{\pi}{2} \left[ \frac{\pi}{2} (\cos x + \sin x) \cos x \right] \right), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases}$
is continuous at $x = 0$, then $a^2 + b^2$ is equal to
JEE Main
Mathematics
View Answer
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f'(x)m + f''(x) = 0$ in m, has two equal roots for every $x \in \mathbb{R}$. If $f(0) = 1, f'(0) = 2$, and $(\alpha, \beta)$ is the largest interval in which the function $f(\log_e x - x)$ is increasing, then $\alpha + \beta$ is equal to ____________.
JEE Main
Mathematics
View Answer

Need Expert Advice?

More Questions from JEE Main

Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________

If $\int (\sin x)^{\frac{-11}{2}} (\cos x)^{\frac{-5}{2}} dx = - \frac{p_1}{q_1} (\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2} (\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3} (\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4} (\cot x)^{\frac{-3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i) = 1$ for $i = 1, 2, 3, 4$ and C is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to _________

The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by,
$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(where x, t and other values have S.I. units). The dielectric constant of the medium is _________.
(speed of light in free space is $3 \times 10^8 \text{ m/s}$)

Two loudspeakers ($L_1$ and $L_2$) are placed with a separation of 10 m, as shown in figure. Both speakers are fed with an audio input signal of same frequency with constant volume. A voice recorder, initially at point $A$, at equidistance to both loud speakers, is moved by 25 m along the line $AB$ while monitoring the audio signal. The measured signal was found to undergo 10 cycles of minima and maxima during the movement. The frequency of the input signal is _________ Hz
(Speed of sound in air is $324 \text{ m/s} \text{ and } \sqrt{5} = 2.23$)

Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.

Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.

Let $[\cdot]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta$ is equal to ________.

Let a vector $\vec{a} = \sqrt{2}\hat{i} - \hat{j} + \lambda\hat{k}, \lambda > 0$, make an obtuse angle with the vector $\vec{b} = -\lambda^2\hat{i} + 4\sqrt{2}\hat{j} + 4\sqrt{2}\hat{k}$ and an angle $\theta, \frac{\pi}{6} < \theta < \frac{\pi}{2}$, with the positive z-axis. If the set of all possible values of $\lambda$ is $(\alpha, \beta) - \{\gamma\}$, then $\alpha + \beta + \gamma$ is equal to ________.

10)

A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$