1
JEE Advanced 2023 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{d^2 f}{d x^2}(x)>0$ 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 $X_f=1$
2
JEE Advanced 2016 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Let $$f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$$ and $$h:\mathbb{R} \to \mathbb{R}$$ be differentiable functions such that $$f\left( x \right)= {x^3} + 3x + 2,$$ $$g\left( {f\left( x \right)} \right) = x$$ and $$h\left( {g\left( {g\left( x \right)} \right)} \right) = x$$ for all $$x \in R$$. Then
A
$$g'\left( 2 \right) = {1 \over {15}}$$
B
$$h'\left( 1 \right) = 666$$
C
$$h\left( 0 \right) = 16$$
D
$$h\left( {g\left( 3 \right)} \right) = 36$$
3
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that
$$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F'\left( x \right) < 0$$ for all $$x \in \left( {{1 \over 2},3} \right).$$ Let $$f\left( x \right) = xF\left( x \right)$$ for all $$x \in R.$$

The correct statement(s) is (are)

A
$$f'\left( 1 \right) < 0$$
B
$$f\left( 2 \right) < 0$$
C
$$f'\left( x \right) \ne 0$$ for any $$x \in \left( {1,3} \right)$$
D
$$f'\left( x \right) = 0$$ for some $$x \in \left( {1,3} \right)$$
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