New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Application of Derivatives question

2013 · Shift 2 · Q27

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Application of Derivatives
2013 · Shift 2 · Q27

Application of Derivatives question

2013 · Shift 2 · Q27

JEE AdvancedMathematicsApplication of DerivativesMCQ+4 / −1

Let

f:\left[ {0,1} \right] \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''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]

. If the function

{e^{ - x}}f\left( x \right)

assumes its minimum in the interval

\left[ {0,1} \right]

at

x = {1 \over 4}

, which of the following is true?

A
$f'\left( x \right) \lt f\left( x \right),{1 \over 4} \lt x \lt {3 \over 4}$
B
$f'\left( x \right) \gt f\left( x \right),0 \lt x \lt {1 \over 4}$
C
$f'\left( x \right) \lt f\left( x \right),0 \lt x \lt {1 \over 4}$
D
$f'\left( x \right) \lt f\left( x \right),{3 \over 4} \lt x \lt 1$

More from Application of Derivatives

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 · MCQ
The function $f (x) = 2 ∣ x ∣ + ∣ x + 2 ∣ - ∣ ∣ x + 2 ∣ - 2 ∣ x ∣ ∣$ has a local minimum or a local maximum at x =2013 · 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 · 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 · 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 · 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 · MCQ
If $f (x) = \int_{0}^{x} e^{t^{2}} (t - 2) (t - 3) d t$ for all $x \in (0, \infty),$ then2012 · 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 · Numerical