Let S be the set of all twice differentiable functions f from R to R such that f"(x)<0 ∀ × ∈ (−2,2). for f∈S, let Xf be the number of points x∈(−1,1) for which f(2x)=3x. Then which of the following options is/are Correct?

The graph of f is always concave down So it can intersect y=3x2 at a maximum of

Let S be the set of all twice differentiable functions f from R to R such that f"(x)<0 ∀ × ∈ (−2,2). for f∈S, let Xf be the number of points x∈(−1,1) for which f(2x)=3x. Then which of the following options is/are Correct?

The graph of f is always concave down So it can intersect y=3x2 at a maximum of

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Let S be the set of all twice differentiable functions f from R to R such that $f " (x) < 0 \forall \times \in (- 2, 2) .$ for $f \in S,$ let $X_{f}$ be the number of points $x \in (- 1, 1)$ for which $f (2 x) = 3 x .$ Then which of the following options is/are Correct?

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There exists a function $f \in S$ such that $X_{f} = 0 .$

For every function $f \in S,$ we have $X_{f} \leq 2$

There exists a function $f \in S,$ we have $X_{f} > 2$

There does Not exist any function $f \in S$ such that $X_{f} = 1$

answer is A, B.

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The graph of f is always concave down
So it can intersect $y = \frac{3 x}{2}$ at a maximum of

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Let S be the set of all twice differentiable functions f from R to R such that f"(x)<0 ∀ × ∈ (−2,2). for f∈S, let Xf be the number of points x∈(−1,1) for which f(2x)=3x. Then which of the following options is/are Correct?

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Let S be the set of all twice differentiable functions f from R to R such that $f " (x) < 0 \forall \times \in (- 2, 2) .$ for $f \in S,$ let $X_{f}$ be the number of points $x \in (- 1, 1)$ for which $f (2 x) = 3 x .$ Then which of the following options is/are Correct?