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.
Similar Questions
- 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 ______
- 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 ____________.
- The number of points of discontinuity of the function $f (x) = \left[\frac{x^2}{2}\right] - [\sqrt{x}], x \in [0,4]$, where $[.]$denotes the greatest integer function, is ________
- The number of functions $f: \{1, 2, 3, 4\} \rightarrow \{a, b, c\}$, which are not onto, is:
- If $y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1 + \sqrt{1 - x^2}}\right)$, then $\frac{dy}{dx}$ at $x = \frac{\sqrt{3}}{2}$ is equal to:
- Let $f(x) = \lim_{\theta \rightarrow 0} \left(\frac{\cos\pi x - x^{\left(\frac{2}{\theta}\right)}\sin(x - 1)}{1 + x^{\left(\frac{2}{\theta}\right)}(x - 1)}\right)$, $x \in \mathbf{R}$. Consider the following two statements :
(I) $f(x)$ is discontinous at $x = 1$.
(II) $f(x)$ is continous at $x = -1$.
Then, - The sum of all the elements in the range of $f(x) = \text{Sgn}(\sin x) + \text{Sgn}(\cos x) + \text{Sgn}(\tan x) + \text{Sgn}(\cot x)$,
$x \neq \frac{n\pi}{2}, n \in \mathbf{Z}$, where $\text{Sgn}(t) = \begin{cases} 1, & \text{if } t > 0 \\ -1, & \text{if } t < 0 \end{cases}$, is : - Let $f$ be a polynomial function such that $\log_2 (f(x)) = \left( \log_2 \left( 2 + \frac{2}{3} + \frac{2}{9} + \dots \infty \right) \right) \cdot \log_3 \left( 1 + \frac{f(x)}{f\left(\frac{1}{x}\right)} \right), x > 0$ and $f(6) = 37$. Then $\sum_{n=1}^{10} f(n)$ is equal to ________.
- If $f(x) = \begin{cases} \frac{a|x| + x^2 - 2(\sin|x|)(\cos|x|)}{x} & , \ x \neq 0 \\ b & , \ x = 0 \end{cases}$ is continuous at $x = 0$, then $a + b$ is equal to
- If the function $f(x) = \frac{e^x\left(e^{\tan x - x} - 1\right) + \log_e(\sec x + \tan x) - x}{\tan x - x}$ is continuous at $x = 0$, then the value of $f(0)$ is equal to
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 ______
- 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 ____________.
For a real number $y$, consider $[y]$ denotes the greatest integer less than or equal to $y$.
If $f(x) = \frac{\tan(\pi[x-\pi])}{1+[x]^2}$, then- The number of points of discontinuity of the function $f (x) = \left[\frac{x^2}{2}\right] - [\sqrt{x}], x \in [0,4]$, where $[.]$denotes the greatest integer function, is ________
- Consider a function $f(x)$ which has exactly two roots at $x=a$. If $\lim_{x \to a}\left(\frac{\lambda f'(x)}{f(x)} - \frac{1}{x-a}\right) = m \, (\neq 0)$, then the value of $\lambda$ is
