QJEEAdv23Ma8164: JEE Advanced 2023 Mathematics - Class 12 JEEAdv23Ma8164 Solution: Let be the set of all twice differentiable functions from to…

Question

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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?

Accepted Answer

S=

Set of all twice differentiable functions

\mathrm{f}: \mathrm{R} \rightarrow

R

\frac{d^{2} f}{{dx}^{2}} > 0 in (- 1, 1)

Graph '

\mathrm{f}

' is Concave upward.

Number of solutions of

f(x)=x \rightarrow x_f

\Rightarrow

Graph of

y=f(x)

can intersect graph of

y=x

at atmost two points

\Rightarrow 0 \leq x_f \leq 2

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