New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Differentiation question

2023 · Shift 2 · Q24

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Differentiation
2023 · Shift 2 · Q24

Differentiation question

2023 · Shift 2 · Q24

JEE AdvancedMathematicsDifferentiationMultiple correct+4 / −2

Let

S

be the set of all twice differentiable functions

f

from

\mathbb{R}

to

\mathbb{R}

such that

\frac{d^2 f}{d x^2}(x)\gt 0

for all

x \in(-1,1)

. For

f \in S

, let

X_f

be the number of points

x \in(-1,1)

for which

f(x)=x

. Then which of the following statements is(are) true?

A
There exists a function $f \in S$ such that $X_f=0$
B
For every function $f \in S$ , we have $X_f \leq 2$
C
There exists a function $f \in S$ such that $X_f=2$
D
There does NOT exist any function $f$ in $S$ such that $X_f=1$

More from Differentiation

Let $f : R \to R, g : R \to R$ and $h : R \to R$ be differentiable functions such that $f (x) = x^{3} + 3 x + 2, g (f (x)) = x$ and $h (g (g (x))) = x$ …2016 · Multiple correct
Let $F : R \to R$ be a thrice differentiable function. Suppose that $F (1) = 0, F (3) = - 4$ and $F^{'} (x) < 0$ for all $x \in (\frac{1}{2}, 3) .$ Let $f (x) = x F (x)$ …2015 · Multiple correct
Let $f : [0, 2] \to R$ be a function which is continuous on $[0, 2]$ and is differentiable on $f (0) = 1$ with $f (0) = 1$ . Let $F (x) = 0 \int x^{2} f (t) d t$ for $x \in [0, 2]$ …2014 · MCQ
Let $f (θ) = sin (tan^{- 1} (\frac{s i n θ}{c o s 2 θ})),$ where $- \frac{π}{4} < θ < \frac{π}{4} .$ Then the value of $\frac{d}{d ( t a n θ )} (f (θ))$ …2011 · Numerical
Consider the functions defined implicitly by the equation $y^{3} - 3 y + x = 0$ on various intervals in the real line. If $x \in (- \infty, - 2) \cup (2, \infty)$ , the equation implicitly defines a unique real valued differentiable function $y = f (x)$ . If $x \in (- 2, 2)$ …2008 · MCQ
Let $f$ and $g$ be real valued functions defined on interval $(- 1, 1)$ such that $g^{''} (x)$ is continuous, $g (0) e 0. g^{'} (0) = 0$ , $g^{''} (0) e 0$ , and $f (x) = g (x) sin x$ …2008 · MCQ
Consider the function $f : (- \infty, \infty) \to (- \infty, \infty)$ defined by $f (x) = \frac{x ^{2} - a x + 1}{x ^{2} + a x + 1}, 0 < a < 2.$ Which of the following is true?2008 · MCQ
Let $g (x) = lo g f (x)$ , where $f (x)$ is a twice differentiable positive function on (0, $\infty$ ) such that $f (x + 1) = x f (x)$ . Then for N = 1, 2, 3, ..., $g^{''} (N + \frac{1}{2}) - g^{''} (\frac{1}{2}) =$ 2008 · MCQ