Question
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
This question was previously asked in
WBJEE 2026 Physics and Chemistry Question Paper (24-May-2026)
The correct answer is
$f'(x)$ does not exist
To determine whether the derivative of the function \(f(x) = \frac{\tan(\pi[x-\pi])}{1+[x]^2}\) exists for all \(x\), we need to analyze the behavior of the function. Here's a detailed explanation:
- The function involves the floor function \([x]\), which denotes the greatest integer less than or equal to \(x\). This can lead to discontinuities at integer points, as the value of \([x]\) changes abruptly.
- The expression \(\tan(\pi [x-\pi])\) needs careful consideration. Since tangent is discontinuous at odd multiples of \(\frac{\pi}{2}\), the function may exhibit discontinuities.
- For any non-integer \(x\), \([x]\) remains constant across a small neighborhood around that number, so in these cases, the derivative can be considered.
- At integer values of \(x\), \([x]\) changes suddenly, causing a discontinuity in the function. This affects the tangent term and consequently the entire expression, which can result in undefined or non-differentiable points due to jumps in function values.
As a result of the discontinuities at integer values of \(x\) caused by the floor function, \(f'(x)\) does not exist for these points. Therefore, the correct answer is:
\(f'(x)\) does not exist
Was this answer helpful?
Similar Questions
- Given $P(x) = x^4 + ax^3 + bx^2 + cx + d$ such that $x=0$ is the only real root of $P'(x) = 0$. If $P(-1) < P(1)$, then in the interval $[-1, 1]$
- 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
- Which of the following statements is always true?
- Let $A = [a, \infty)$ denotes the domain, then $f: [a, \infty) \to B$, which is defined by $f(x) = 2x^3 - 3x^2 + 6$ will have an inverse for the smallest real value of '$a$' if
- Number of elements in the range set of $f(x) = \left[ \frac{x}{15} \right] \left[ -\frac{15}{x} \right]$, for all $x \in (0, 90)$; (where $[\cdot]$ denotes the greatest integer function) is
- Let domain and range of $f(x)$ and $g(x)$ is $[0, \infty)$. If $f(x)$ is an increasing function, $g(x)$ is a decreasing function, $h(x) = f\{g(x)\}, h(0) = 0$ and $p(x) = h(x^3 - 2x^2 + 2x) - h(4)$, then for all $x \in (0, 2)$
- If the domain of $f(x)$ is $(0, 1)$, then the domain of $y = f(e^x) + f(\ln|x|)$ is
- A figure is bounded by the curves $y = x^2 + 1, y = 0, x = 0$ and $x = 1$. The point at which a tangent should be drawn to the curve $y = x^2 + 1$ for it to cut off trapezium of the greatest area from the figure is
- Let $f(x)$ be a twice differentiable function in $[1, 3]$ and $f(1) = f(3)$. Further if $|f''(x)| \le 2$, then for all $x$ in $[1, 3]$
- If $f$ be a real valued function defined for all real numbers $x$ such that for some fixed $a > 0$, it satisfies $f(x+a) = \frac{1}{2} + \sqrt{f(x) - (f(x))^2} \, \forall x$, then $f(x)$ is periodic with period
Important Questions from Differential Calculus
- Let the domain of the function $f(x) = \log_3 \log_5 \left(7 - \log_2 \left(x^2 - 10x + 85\right)\right) + \sin^{-1} \left(\left|\frac{3x - 7}{17 - x}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha + \beta$ is equal to :
- Let $[\cdot]$ denote the greatest integer function, and let $f(x) = \min \{\sqrt{2}x, x^2\}$.
Let $S = \{x \in (-2, 2) : \text{the function } g(x) = |x|[x^2] \text{ is discontinuous at } x\}$.
Then $\sum_{x \in S} f(x)$ equals - 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
- 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 ______
- 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.
Need Expert Advice?
More Questions from WBJEE
