The function \[f(x)=1-{{e}^{-{{x}^{2}}/2}}\] is [AMU 1999]

The function \[{{x}^{x}}\] is increasing, when [MP PET 2003]

A ball thrown vertically upwards falls back on the ground after 6 second. Assuming that the equation of motion is of the form \[s=ut-4.9{{t}^{2}}\], where s is in metre and t is in second, find the velocity at \[t=0\]

If the path of a moving point is the curve \[x=at\], \[y=b\sin at\], then its acceleration at any instant [SCRA 1996]

If \[\sqrt{(1-{{x}^{6}})}+\sqrt{(1-{{y}^{6}})}={{a}^{3}}({{x}^{3}}-{{y}^{3}})\], then \[\frac{dy}{dx}=\] [Roorkee 1994]

If \[y={{\sec }^{-1}}\frac{2x}{1+{{x}^{2}}}+{{\sin }^{-1}}\frac{x-1}{x+1}\],then \[\frac{dy}{dx}\]is equal to [Pb. CET 2000]

The derivative of \[{{\tan }^{-1}}\left( \frac{\sqrt{1+{{x}^{2}}}-1}{x} \right)\]with respect to \[{{\tan }^{-1}}\left( \frac{2x\sqrt{1-{{x}^{2}}}}{1-2{{x}^{2}}} \right)\]at \[x=0\], is

The values of ?a? for which the function \[(a+2){{x}^{3}}-3a{{x}^{2}}+9ax-1\] decreases monotonically throughout for all real x, are [Kurukshetra CEE 2002]

If a spherical balloon has a variable diameter \[3x+\frac{9}{2}\], then the rate of change of its volume with respect to x is

At what points of the curve \[y=\frac{2}{3}{{x}^{3}}+\frac{1}{2}{{x}^{2}},\]tangent makes the equal angle with axis [UPSEAT 1999]

If the normal to the curve \[y=f(x)\] at the point \[(3,\,4)\] makes an angle \[\frac{3\pi }{4}\]with the positive x-axis then \[f'(3)\] is equal to [IIT Screening 2000; DCE 2001]

In [0, 1] Lagrange's mean value theorem is NOT applicable to [IIT Screening 2003]

If the function \[f(x)={{x}^{3}}-6a{{x}^{2}}+5x\]satisfies the conditions of Lagrange's mean value theorem for the interval [1, 2] and the tangent to the curve \[y=f(x)\]at \[x=\frac{7}{4}\]is parallel to the chord that joins the points of intersection of the curve with the ordinates \[x=1\] and \[x=2\]. Then the value of \[a\]is [MP PET 1998]

Let \[f(x)=\left\{ \begin{align} & {{x}^{\alpha }}\ln x,x>0 \\ & 0,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right\}\], Rolle?s theorem is applicable to f for \[x\in [0,1]\], if \[\alpha =\] [IIT Screening 2004]

The value of the function \[(x-1){{(x-2)}^{2}}\] at its maxima is