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Application of Derivatives

33 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Application of Derivatives

Application of Derivatives

33 questions · Mathematics · JEE Advanced

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be defined by $f (x) = ⎩ ⎨ ⎧ \frac{6 x + sin x}{2 x + sin x}, \frac{7}{3}, if x e q 0, if x = 0.$ Then which of the following…2025 · Shift 2 · Q24 · Multiple correct
Let $Q$ be the cube with the set of vertices ${(x_{1}, x_{2}, x_{3}) \in R^{3} : x_{1}, x_{2}, x_{3} \in {0, 1}}$ . Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$ . Let $S$ …2023 · Shift 1 · Q22 · MCQ
Let $α = k = 1 \sum \infty sin^{2 k} (\frac{π}{6})$ Let $g : [0, 1] \to R$ be the function defined by $g (x) = 2^{α x} + 2^{α (1 - x)} .$ Then, which of the following statements is/are…2022 · Shift 2 · Q28 · Multiple correct
Consider the rectangles lying the region ${(x, y) \in R \times R : 0 \leq x \leq \frac{π}{2}$ and $0 \leq y \leq 2 sin (2 x)}$ and having one side on the X-axis. The area of the rectangle…2020 · Shift 1 · Q24 · MCQ
Let f : R $\to$ R be given by $f (x) = (x - 1) (x - 2) (x - 5)$ . Define $F (x) = 0 \int x f (t) d t$ , x > 0 Then which of the following options is/are correct?2019 · Shift 2 · Q20 · Multiple correct
Let, $f (x) = \frac{s i n π x}{x ^{2}}$ , x > 0 Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f. Then…2019 · Shift 2 · Q21 · Multiple correct
For each positive integer n, let $y_{n} = \frac{1}{n} (n + 1) (n + 2) ... (n + n)^{\frac{1}{n}}$ . For x $\in$ R, let [x] be the greatest integer less than or equal to x. If $n \to \infty lim y_{n} = L$ , then the value…2018 · Shift 1 · Q29 · Numerical
By approximately matching the information given in the three columns of the following table. Let f(x) = x + loge x $-$ x loge x, x $\in$ (0, $\infty$ ) Column 1 contains information about zeroes of f(x), f'(x) and f"(x). Column 2 contains… Includes table2017 · Shift 1 · Q34 · MCQ
By approximately matching the information given in the three columns of the following table. Let f(x) = x + loge x $-$ x loge x, x $\in$ (0, $\infty$ ) Column 1 contains information about zeroes of f(x), f'(x) and f"(x). Column 2 contains… Includes table2017 · Shift 1 · Q35 · MCQ
By approximately matching the information given in the three columns of the following table. Let f(x) = x + loge x $-$ x loge x, x $\in$ (0, $\infty$ ) Column 1 contains information about zeroes of f(x), f'(x) and f"(x). Column 2 contains… Includes table2017 · Shift 1 · Q36 · MCQ
f : R $\to$ R is a differentiable function such that f'(x) > 2f(x) for all x $\in$ R, and f(0) = 1 then2017 · Shift 2 · Q26 · Multiple correct
If $f (x) = cos 2 x - cos x sin x cos 2 x cos x sin x sin 2 x - sin x cos x$ , then2017 · Shift 2 · Q32 · Multiple correct
The least value of a $4 a x^{2} + \frac{1}{x} \geq 1,$ for which $4 a x^{2} + \frac{1}{x} \geq 1,$ , for all $x > 0$ . is2016 · Shift 1 · Q24 · MCQ
Let f: R $\to (0, \infty)$ and g : R $\to$ R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f' $(2) =$ g $(2) = 0$ , f'' $(2) e 0$ and g' $(2) e 0$ . If $x \to 2 lim \frac{f ( x ) g ( x )}{f ^{'} ( x ) g ^{'} ( x )} = 1,$ …2016 · Shift 2 · Q28 · Multiple correct
A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V m m^{3}$ , has a $2$ mm thick solid wall and is open at the top. The bottom of the container is a solid…2015 · Shift 1 · Q27 · Numerical
Let $f, g :$ $[- 1, 2] \to R$ be continuous functions which are twice differentiable on the interval $(- 1, 2)$ . Let the values of f and g at the points $- 1, 0$ and $2$ be as given in the following table: In each of the… Includes table2015 · Shift 2 · Q33 · Multiple correct
The slope of the tangent to the curve $(y - x^{5})^{2} = x (1 + x^{2})^{2}$ at the point $(1, 3)$ is2014 · Shift 1 · Q27 · Numerical
A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8 : 15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed…2013 · Shift 1 · Q32 · Multiple correct
Let $f : [0, 1] \to R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f (0) = f (1) = 0$ and satisfies $f^{''} (x) - 2 f^{'} (x) + f (x) \geq . e^{x}, x \in [0, 1]$ …2013 · Shift 2 · Q27 · MCQ
Let $f : [0, 1] \to R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f (0) = f (1) = 0$ and satisfies $f^{''} (x) - 2 f^{'} (x) + f (x) \geq . e^{x}, x \in [0, 1]$ …2013 · Shift 2 · Q28 · MCQ
The function $f (x) = 2 ∣ x ∣ + ∣ x + 2 ∣ - ∣ ∣ x + 2 ∣ - 2 ∣ x ∣ ∣$ has a local minimum or a local maximum at x =2013 · Shift 2 · Q40 · Multiple correct
Let $f : I R \to I R$ be defined as $f (x) = ∣ x ∣ + x^{2} - 1 .$ The total number of points at which $f$ attains either a local maximum or a local minimum is2012 · Shift 1 · Q30 · Numerical
Let $p (x)$ be a real polynomial of least degree which has a local maximum at $x = 1$ and a local minimum at $x = 3$ . If $p (1) = 6$ and $p (3) = 2$ , then $p^{'} (0)$ is2012 · Shift 1 · Q31 · Numerical
Let $f (x) = (1 - x)^{2} sin^{2} x + x^{2}$ for all $x \in I R$ and let $g (x) = 1 \int x (\frac{2 ( t - 1 )}{t + 1} - I n t) f (t) d t$ for…2012 · Shift 2 · Q28 · MCQ
Let $f (x) = (1 - x)^{2} sin^{2} x + x^{2}$ for all $x \in I R$ and let $g (x) = 1 \int x (\frac{2 ( t - 1 )}{t + 1} - I n t) f (t) d t$ for…2012 · Shift 2 · Q29 · MCQ
If $f (x) = \int_{0}^{x} e^{t^{2}} (t - 2) (t - 3) d t$ for all $x \in (0, \infty),$ then2012 · Shift 2 · Q30 · Multiple correct
Let $f$ be a real-valued differentiable function on $R$ (the set of all real numbers) such that $f (1) = 1$ . If the $y$ -intercept of the tangent at any point $P (x, y)$ on the curve $y = f (x)$ is equal to the cube of the abscissa of $P$ , then find…2010 · Shift 1 · Q40 · Numerical
Let $f$ be a function defined on $R$ (the set of all real numbers) such that $f^{'} (x) = 2010 (x - 2009) (x - 2010)^{2} (x - 2011)^{3} (x - 2012)^{4}$ for all $x \in R$ If $g$ …2010 · Shift 2 · Q22 · Numerical
The maximum value of the function $f (x) = 2 x^{3} - 15 x^{2} + 36 x - 48$ on the set $A = {x ∣ x^{2} + 20 \leq 9 x ∣}$ is .2009 · Shift 2 · Q26 · Numerical
Let $p (x)$ be a polynomial of degree $4$ having extremum at $x = 1, 2$ and $x \to 0 lim (1 + \frac{p ( x )}{x ^{2}}) = 2$ . Then the value of $p (2)$ is2009 · Shift 2 · Q27 · Numerical
For the function $f (x) = x cos \frac{1}{x}, x \geq 1,$ 2009 · Shift 2 · Q28 · Multiple correct
The total number of local maxima and local minima of the function $f (x) = {(2 + x)^{3}, x^{2/3}, - 3 < x \leq - 1 - 1 < x < 2$ is2008 · Shift 1 · Q45 · MCQ
Let $f (x)$ be differentiable on the interval (0, $\infty$ ) such that $f (1) = 1$ , and $t \to x lim \frac{t ^{2} f ( x ) - x ^{2} f ( t )}{t - x} = 1$ for each $x > 0$ . Then $f (x)$ is2007 · Shift 1 · Q24 · MCQ