The interval, inwhich the function f(x) = x²e^-x is an increasing function, will be

• $\left(- \infty , \infty \right)$

• (- 2, 0)

• $\left(2, \infty \right)$

• (0, 2)

The minimum distance from the point (4, 2) to the parabola y² = 8x is

• $\sqrt{2}$

• $2\sqrt{2}$

• 2

• $3\sqrt{2}$

A particle moves so that the space described in time t is square root of a quadratic function of t. Then,

• $\mathrm{v} \propto \left(\frac{1}{\mathrm{s}}\right)$

• $\mathrm{acceleration} \propto \frac{1}{{\mathrm{s}}^3}$

• $\mathrm{acceleration} \propto {\mathrm{s}}^3$

• None of these

The points at which the tangent to the curve y = x³ + 5 is perpendicular to the line x + 3y = 2 are

• (6, 1), (- 1, 4)

• (1, 6), (1, 4)

• (6, 1), (4, - 1)

• (1, 6), (- 1, 4)

Let f(x) = a - (x - 3)^8/9, then maxima of f(x) is

• 3

• a - 3

• a

• None of these

If the total cost C(x) in rupees associated with the production of x units of an item is given by C (x) = 3x³ - 2x² + x + 100. Then, the marginal change in cost, when x = 5, is

• 200

• 225

• 206

• 226

If f'(x) > 0, $\mathrm{x} \in \mathrm{R}, \mathrm{f}\text{'}\left(3\right) = 0$ and g(x) = $\mathrm{f}\left({\tan }^2\left(\mathrm{x}\right) - 2\tan \left(\mathrm{x}\right) + 4\right), 0 < \mathrm{x} < \frac{\mathrm{\pi }}{2}$, then g(x) is increasing in

• $\left(0, \frac{\mathrm{\pi }}{4}\right)$

• $\left(\frac{\mathrm{\pi }}{6}, \frac{\mathrm{\pi }}{3}\right)$

• $\left(0, \frac{\mathrm{\pi }}{3}\right)$

• $\left(\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right)$

The radius of a cylinder is increasing at the rate of 2 m/s and its height is decreasing at the rate of 3 m/s. When the radius is 3 m and height is 5 m, then the volume of the cylinder would change at the rate of

• $87\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $33\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $27\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

• $15\mathrm{\pi } {\mathrm{m}}^3/\mathrm{s}$

The values of a, if f(x) =$2{\mathrm{e}}^{\mathrm{x}} - {\mathrm{ae}}^{-\mathrm{x}} + \left(2\mathrm{a} +\hspace{0.17em}1\right)\mathrm{x} - 3$  increases x, are in

• $[0, \infty )$

• $(- \infty , 0]$

• $\left(- \infty , \infty \right)$

• $\left(1, \infty \right)$

A cylindrical tank of radius 2 m is being filled with rice at the rate of 314 cubic m/h. The depth ofthe rice is increasing at the rate of

• 25 cubic m/h

• 0.25 cubic m/h

• 1 cubic m/h

• $\frac{3}{4}$ cubic m/h