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Differentiation

9 questions · Mathematics · JEE Advanced

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PYQ Library
JEE Advanced
Mathematics
Differentiation

Differentiation

9 questions · Mathematics · JEE Advanced

Let $S$ be the set of all twice differentiable functions $f$ from $R$ to $R$ 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…2023 · Shift 2 · Q24 · Multiple correct
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 · Shift 1 · Q26 · 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 · Shift 2 · Q38 · Multiple correct
Let $f : [0, 2] \to R$ be a function which is continuous on $[0, 2]$ and is differentiable on $(0, 2)$ with $f (0) = 1$ . Let $F (x) = 0 \int x^{2} f (t) d t$ for $x \in [0, 2]$ …2014 · Shift 2 · Q34 · 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 · Shift 1 · Q33 · 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 · Shift 1 · Q31 · 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 · Shift 1 · Q42 · 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 · Shift 2 · Q31 · 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 · Shift 2 · Q35 · MCQ