If $$y(x)=x^{x},x > 0$$, then $$y''(2)-2y'(2)$$ is equal to

Let $$f(x) = 2x + {\tan ^{ - 1}}x$$ and $$g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$$. Then

Let $$y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$$. ...

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7...

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is :

Suppose $$f: \mathbb{R} \rightarrow(0, \infty)$$ be a differentiable function such that $$5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}$$. If ...

Let $$f$$ and $$g$$ be the twice differentiable functions on $$\mathbb{R}$$ such that $$f''(x)=g''(x)+6x$$ $$f'(1)=4g'(1)-3=9$$ $$f(2)=3g(2)=12$$. The...

Let $$y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})$$. Then $$y' - y''$$ at $$x = - 1$$ is equal to

If $$f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$$, then

Let $$x(t)=2 \sqrt{2} \cos t \sqrt{\sin 2 t}$$ and $$y(t)=2 \sqrt{2} \sin t \sqrt{\sin 2 t}, t \in\left(0, \frac{\pi}{2}\right)$$. Then $$\frac{1+\lef...

The value of $$\log _{e} 2 \frac{d}{d x}\left(\log _{\cos x} \operatorname{cosec} x\right)$$ at $$x=\frac{\pi}{4}$$ is

Let f : R $$\to$$ R be a function defined by f(x) = (x $$-$$ 3)n1 (x $$-$$ 5)n2, n1, n2 $$\in$$ N. Then, which of the following is NOT true?...

If $${\cos ^{ - 1}}\left( {{y \over 2}} \right) = {\log _e}{\left( {{x \over 5}} \right)^5},\,|y|

If $$y = {\tan ^{ - 1}}\left( {\sec {x^3} - \tan {x^3}} \right),{\pi \over 2}

The function $$f(x) = {x^3} - 6{x^2} + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements :Statement 1 : there exists x1, x2 $$\in$$(2...

If $$y(x) = {\cot ^{ - 1}}\left( {{{\sqrt {1 + \sin x} + \sqrt {1 - \sin x} } \over {\sqrt {1 + \sin x} - \sqrt {1 - \sin x} }}} \right),x \in \left...

The local maximum value of the function $$f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$$, x > 0, is

Let $$A = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {...

If Rolle's theorem holds for the function $$f(x) = {x^3} - a{x^2} + bx - 4$$, $$x \in [1,2]$$ with $$f'\left( {{4 \over 3}} \right) = 0$$, then ordere...

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'(x) $$ \ne $$ 0 for all x $$ \in $$ R. If $$\left| {\matrix{...

The derivative of $${\tan ^{ - 1}}\left( {{{\sqrt {1 + {x^2}} - 1} \over x}} \right)$$ with respect to $${\tan ^{ - 1}}\left( {{{2x\sqrt {1 - {x^2}}...

If the function $$f\left( x \right) = \left\{ {\matrix{ {{k_1}{{\left( {x - \pi } \right)}^2} - 1,} & {x \le \pi } \cr {{k_2}\cos x,} &...

If $$\left( {a + \sqrt 2 b\cos x} \right)\left( {a - \sqrt 2 b\cos y} \right) = {a^2} - {b^2}$$ where a > b > 0, then $${{dx} \over {dy}}\,\,at...

If y2 + loge (cos2x) = y, $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$$, then :

If $$x = 2\sin \theta - \sin 2\theta $$ and $$y = 2\cos \theta - \cos 2\theta $$, $$\theta \in \left[ {0,2\pi } \right]$$, then $${{{d^2}y} \over {...

Let ƒ and g be differentiable functions on R such that fog is the identity function. If for some a, b $$ \in $$ R, g'(a) = 5 and g(a) = b, then ƒ'(b) ...

If c is a point at which Rolle's theorem holds for the function, f(x) = $${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$$ in the interval [...

Let y = y(x) be a function of x satisfying $$y\sqrt {1 - {x^2}} = k - x\sqrt {1 - {y^2}} $$ where k is a constant and $$y\left( {{1 \over 2}} \right...

If $$y\left( \alpha \right) = \sqrt {2\left( {{{\tan \alpha + \cot \alpha } \over {1 + {{\tan }^2}\alpha }}} \right) + {1 \over {{{\sin }^2}\alpha }...

The derivative of $${\tan ^{ - 1}}\left( {{{\sin x - \cos x} \over {\sin x + \cos x}}} \right)$$, with respect to $${x \over 2}$$ , where $$\left( {x ...

If ey + xy = e, the ordered pair $$\left( {{{dy} \over {dx}},{{{d^2}y} \over {d{x^2}}}} \right)$$ at x = 0 is equal to :

If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is :

If $$2y = {\left( {{{\cot }^{ - 1}}\left( {{{\sqrt 3 \cos x + \sin x} \over {\cos x - \sqrt 3 \sin x}}} \right)} \right)^2}$$, x $$ \in $$ $$\left( {...

For x > 1, if (2x)2y = 4e2x$$-$$2y, then (1 + loge 2x)2 $${{dy} \over {dx}}$$ is equal to : ...

If   x $$=$$ 3 tan t and y $$=$$ 3 sec t, then the value of $${{{d^2}y} \over {d{x^2}}}$$ at t $$ = {\pi \over 4},$$ is :

Let f be a polynomial function such that f (3x) = f ' (x) . f '' (x), for all x $$ \in $$ R. Then :

If $$g$$ is the inverse of a function $$f$$ and $$f'\left( x \right) = {1 \over {1 + {x^5}}},$$ then $$g'\left( x \right)$$ is equal to:

If $$x=-1$$ and $$x=2$$ are extreme points of $$f\left( x \right) = \alpha \,\log \left| x \right|+\beta {x^2} + x$$ then

If $$y = \sec \left( {{{\tan }^{ - 1}}x} \right),$$ then $${{{dy} \over {dx}}}$$ at $$x=1$$ is equal to :

$${{{d^2}x} \over {d{y^2}}}$$ equals:

Let $$f:\left( { - 1,1} \right) \to R$$ be a differentiable function with $$f\left( 0 \right) = - 1$$ and $$f'\left( 0 \right) = 1$$. Let $$g\left( ...

Let $$y$$ be an implicit function of $$x$$ defined by $${x^{2x}} - 2{x^x}\cot \,y - 1 = 0$$. Then $$y'(1)$$ equals

The set of points where $$f\left( x \right) = {x \over {1 + \left| x \right|}}$$ is differentiable is

If $${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$ then $${{{dy} \over {dx}}}$$ is

Let $$f:R \to R$$ be a differentiable function having $$f\left( 2 \right) = 6$$, $$f'\left( 2 \right) = \left( {{1 \over {48}}} \right)$$. Then $$...

If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals

The value of $$a$$ for which the sum of the squares of the roots of the equation $${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least v...

If $$x = {e^{y + {e^y} + {e^{y + .....\infty }}}}$$ , $$x > 0,$$ then $${{{dy} \over {dx}}}$$ is

If $$f\left( y \right) = {e^y},$$ $$g\left( y \right) = y;y > 0$$ and $$F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \rig...

If $$f\left( x \right) = {x^n},$$ then the value of $$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{...

Let $$f\left( x \right)$$ be a polynomial function of second degree. If $$f\left( 1 \right) = f\left( { - 1} \right)$$ and $$a,b,c$$ are in $$A.P, $$...

If $$y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},$$ then $$\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}$$ is

If $$f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)$$ and $$g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x)$$, then the value of $$f(4)-g(4)$$ i...

Let $$f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}$$ For $$\mathrm{n} \geq 2$$, define $$f^{\mathrm{n}}(x)=f^{1} \mathrm...

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a differentiable function that satisfies the relation $$f(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}$$. If $$f'(0)...

Let $$f:[0,1] \rightarrow \mathbf{R}$$ be a twice differentiable function in $$(0,1)$$ such that $$f(0)=3$$ and $$f(1)=5$$. If the line $$y=2 x+3$$ in...

Let $$f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^...

Let f and g be twice differentiable even functions on ($$-$$2, 2) such that $$f\left( {{1 \over 4}} \right) = 0$$, $$f\left( {{1 \over 2}} \right) = 0...

If $$y(x) = {\left( {{x^x}} \right)^x},\,x > 0$$, then $${{{d^2}x} \over {d{y^2}}} + 20$$ at x = 1 is equal to ____________.

Let f(x) be a cubic polynomial with f(1) = $$-$$10, f($$-$$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $$-$$1. Then f...

If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the ...

If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then $${{{d^2}y} \over {d{x^2}}}$$ at x = 0 is equal to ___________.

If $$f(x) = \sin \left( {{{\cos }^{ - 1}}\left( {{{1 - {2^{2x}}} \over {1 + {2^{2x}}}}} \right)} \right)$$ and its first derivative with respect to x ...

If y = $$\sum\limits_{k = 1}^6 {k{{\cos }^{ - 1}}\left\{ {{3 \over 5}\cos kx - {4 \over 5}\sin kx} \right\}} $$, then $${{dy} \over {dx}}$$ at x = 0 i...