Continuity Questions - JEE

Let $f$ be defined for all $x \in \mathbb{R}$. If $f$ is differentiable and $f(x^3) = x^5$ for all $x \in \mathbb{R}\,(x\ne0)$, then $f'(27)$ is equal to:

If $y = f(x)$ satisfies $f(x+y) = f(x)+f(y)+2xy-1$ for all $x,y\in\mathbb{R}$ and $f'(0) = \cos\alpha$, then which of the following is/are true?

$h(x)=\min\{\{x\},1-\{x\}\}$. h is:

Let $f$, $g$ and $h$ be differentiable functions. If $f(0) = 1$, $g(0) = 2$, $h(0) = 3$ and the derivative of their pairwise product at $x = 0$ are $(fg)'(0) = 6$, $(gh)'(0) = 4$ and $(hf)'(0) = 5$, then compute the value of $(fgh)'(0)$.

771. Let $f: R \to R$ be a function defined by $f(x) = \begin{cases} x^3 + 2x^2 + x + c, & \text{if } x \leq b \\ e^x, & \text{if } x > b \end{cases}$, where $b$ and $c$ are integers. If $f(x)$ is differentiable $\forall\, x \in R$, then find the value of $(b + c)$.

Let $f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ where $a, b, c, d$ are in A.P. If $f(x)$ is continuous, then the number of local minima of $y = |f(|x|)|$ is:

Let \(g(x) = \begin{cases} k\sqrt{x+1}; & 0 \leq x \leq 3 \\ mx + 2; & 3

A certain function $f(x)$ has the property that $f(3x) = af(x)$ for all positive real values of $x$ and $f(x) = 1 - |x - 2|$ for $1 \le x \le 3$. Find $\lim_{x \to 2} (f(x))^{\frac{\csc\left(\frac{\pi x}{2}\right)}{1}}$

If $f(x) = \begin{cases} x\tan^{-1}(1/x), & x \in (-\infty,-1]\cup[1,\infty) \\ 0, & x = 0 \end{cases}$, then $f'(0)$ is:

If \(\alpha,\ \beta,\ (\alpha

Question 23: Given We have f: (-1, 1) → ℝf(0) = -1, f'(0) = 1If g(x) = [f(2f(x) + 2)]^(1/2)Find g'(0)

The value of limx→0(1 + sin x)cos x is:

Evaluate: $\lim_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac{\sin x}{x - \sin x}}$

If $f(x) = \sin\!\left[\tan^{-1}\!\left(\dfrac{1-x^2/2}{x}\right)\right]$, $g(x) = \cos\!\left[2\tan^{-1}\!\left(\dfrac{x}{1+\sqrt{1-x^2}}\right)\right]$, and $h(x)=f(x)-g(x)$, then $h'(1/2)$:

\(f(x)=\begin{cases}|x-3| & x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} & x is:

limx→0 cos(sin x)−cos x x4 is equal to:

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a),\, f'(b),\, f'(c)$ are in:

If $f(\theta) = \tan^{-1}\!\left(\dfrac{\sin\theta-\cos\theta}{\sin\theta+\cos\theta}\right)$, then which of the following are correct?

If limx→0 x2 sin βx αx−sin x = 1, find 6(α + β).

Let $y(x) = (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $\dfrac{d}{dx}\!\big[y(x)\big]_{x=1}$ equals: [Integer type]

Let $|f(x)-f(y)|\le 2|x-y|^{3/2}$, $f(0)=1$. Find $\int_0^1 f^2(x)\,dx$.

Given $f$ piecewise defined. $f_1(x)=|f(|x|)|$. If $f$ is continuous in $[-2,10]$, then:

The value of limx→0 | cos(sin 3x)|−1 x2 is:

Evaluate: $\lim_{x \to 0} \left(\tan \frac{x}{x}\right)^{\frac{1}{x}}$

Let $f(x)$ be a polynomial function of second degree. If $f(1) = f(-1)$ and $a, b, c$ are in A.P., then $f'(a)$, $f'(b)$, $f'(c)$ are in:

Suppose $f$ is a derivable function that satisfies the equation $ f(x+y) = f(x) + f(y) + x^2y + xy^2 $ for all real numbers $x$ and $y$. Suppose that $ \lim_{x \to 0}\left[\dfrac{f(x)}{x}\right] = 1 $, find $ f(3) = $ __________.

The value of $\lim_{x \to 0} \frac{(\tan x - \sin x)}{(\tan x - \sin x) + (\tan x + \sin x)} + \frac{1}{r^3 - r}$ is

882. Let $f(x) = \begin{cases} \dfrac{ax^3 + bx^2 + cx + d}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$ be a continuous function where $a, b, c, d$ are in arithmetic progression. Then find the number of points where $|f(|x|)|$ is non derivable.

The value of $\lim_{n \to \infty} \sqrt[n]{a^n + b^n}$ (where $a > 0, b > 0$) is equal to

If $y = e^{\sin^{-1}x} + e^{\cos^{-1}x}$, then which of the following is/are true?

Let fn(x) = limt→x $f_n(x) = \lim_{t \to x} \frac{\sin^{-1}(nt)}{2t} = \frac{\sin^{-1}(nx)}{2x}$. Find the value of $\lim_{x \to 0} \left[\frac{\sin^2 2x}{2x}\right] + \left[\frac{\sin^{-1} 4x}{2x}\right]$.

279. Let $ f(x) = \dfrac{x \ln x - \ln x}{9x^2 - 2e^x x - 9x + 2e^x} + 2 $ and $ g(x) = \sin^2\left(\dfrac{\pi x^2}{2}\right) $, then the value of $ \lim_{x \to 1} \dfrac{f(x)}{g(x)} $ is:

For the curve \sin x + \sin y = 1 lying in the first quadrant, there exists a constant a for which \lim_{x \to 0} x^a \frac{d^2y}{dx^2} = L (not zero). Find 2a.

Evaluate: $\lim_{x \to \infty} \left(\frac{3x^2 + 1}{4x^2 - 1}\right)^{\frac{x^2}{1+x}}$

Let $ f(x) = \cot^{-1}\left(\text{sgn}\left(\dfrac{[x]}{2x - [x]}\right)\right) $:Statement-1: $ f(x) $ is discontinuous at $ x = 1 $.Statement-2: $ f(x) $ is non-differentiable at $ x = 1 $.Which of the following option is correct?[Note: $[k]$ denotes greatest integer function less than or equal to $k$.]

Let a = min{x2 + 2x + 3, x ∈R} and b = limθ→0 1−cos θ θ2 . Then the value of Pn r=0 arbn−r is:

If $x = \sec\theta - \cos\theta$ and $y = \sec^n\theta - \cos^n\theta$, then $\left(x^2+4\right)\!\left(\dfrac{dy}{dx}\right)^{\!2}$ equals $n^2(y^2+k)$. Find $k$.

If $f(x) = \frac{2\sin x \cos x + 2x \cot x}{2\sin x}$, find $f'\left(\frac{\pi}{4}\right)$ and verify it equals $\sqrt{2}$

Let $f(x) = \sin^2(\sin x)$. If $g$ is the inverse of $f$, find $g''(3)$.Given: $f'(0) = \sin^2(\sin 1)$, $f''(0) = 2\sin(\sin 1)\cos(\sin 1)\cos 1$\[g''(y) = \frac{-1}{[f'(x)]^3} f''(x)\]

Let $y = \cot^{-1}(1) + \cot^{-1}(2) + \cot^{-1}(3) + \cdots$. If $\dfrac{d}{dn}\!\left[\sum_{r=1}^{n}\cot^{-1}(r)\right]$ is evaluated and the sum $\sum_{r=1}^{10}\cot^{-1}(r^2-r+1)$ equals $\tan^{-1}(k/l)$, find $10k+l$ (answer 101 from key).

If $y^2 + \ln(\cos^2 x) = y$, then $|y''(0)+y'(0)|$ equals: [Integer type]

If $f(x) = \begin{cases} \frac{\tan^2\{x\}}{x}, & x > 0 \\ 1, & x = 0 \\ \{x\}\cot\{x\}, & x where \([x]$ is the integral part of $x$ and $\{x\}$ is the fractional part of $x$, then

Let $f:(−\infty,\infty)−\{0\}\to\mathbb{R}$ be a differentiable function such that $f'(1)=\displaystyle\lim_{a\to\infty}a^2f\!\left(\frac{1}{a}\right)$. Then $\displaystyle\lim_{a\to\infty}\frac{a(a+1)}{2}\tan^{-1}\!\left(\frac{1}{a}\right)+a^2-2\log_e a$ is equal to

For each $ x \in \mathbb{R} $, let $ [x] $ be the greatest integer less than or equal to x. Then $ \lim_{x \to 0} \frac{x[|x|] \sin[x]}{|x|} $ is equal to

For each $ t \in \mathbb{R} $, let $ [t] $ be the greatest integer less than or equal to t. Then $ \lim_{x \to 1^-} \frac{(1-|x|+\sin[1-x])\sin\left(\frac{\pi}{2}[1-x]\right)}{|1-x|[|1-x|]} $

If $\lim_{x \to -1} \frac{x^2 - ax + b}{x - 1} = 5$, then $a + b$ is equal to :-

\[ \lim_{x \to 2} \frac{\sqrt{1 - \cos{2(x - 2)}}}{x - 2} \]

If $ \lim_{x \to 2} \frac{\tan{(x - 2)}{(x^2 + (k - 2)x - 2k)}}{x^2 - 4x + 4} = 5 $ then k is equal to

Given: $x^2 + y^2 + \sin y = 4$. Find $-\dfrac{d^2y}{dx^2}\bigg|_{(-2,0)}$.

If $f(x) = \log_x(\log_e x)$, then $f'(e)$ equals:

Continuity Questions (1086)