Application of Derivatives

The functions f(x) =xe ^{- X}, $\forall \left(\mathrm{x} \in \mathrm{R}\right)$ attains a maximum value at x is equal

• 1

• 2

• $\frac{1}{\mathrm{e}}$

• 3

The approximate value of ${\left(1.0002\right)}^{3000} \mathrm{is}$

• $1·2$

• $1·4$

• $1·6$

• $1·8$

The sum of two numbers is 20. If the product of the square of one number and cube of the other is maximum, then the numbers are :

• 12, 8

• 3, 4

• 9, 12

• 15, 18

Gas is being pumped into a spherical balloon at the rate of 30 ft³/min. The rate at which the radius increase when it reaches the value 15ft, is :

• $\frac{1}{30\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{15\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{20\mathrm{\pi }} \mathrm{ft}/\min$

• $\frac{1}{25\mathrm{\pi }} \mathrm{ft}/\min$

The minimum value of 2x² + x - 1 is :

• $- \frac{1}{4}$

• $\frac{3}{2}$

• $- \frac{9}{8}$

• $\frac{9}{8}$

A point is moving on y = 4 - 2x². The x-coordinate of the point is decreasing at the rate of 5 units/second. Then, the rate at which y coordinate of the point is changing when the point is at (1, 2) is

• 5 units/s

• 10 units/s

• 15 units/s

• 20 units/s

Match the points on the curve 2y² = x + 1 with the slopes of normals at those points and choose the correct answer.

      Point           Slope of the normal

A    (7, 2)         1 - 4$\sqrt{2}$

B    (0, $\frac{1}{\sqrt{2}}$)     2  - 8

C    (1, - 1)       3, 4

D    (3, $\sqrt{2}$)      4, 0

$\mathrm{f}\left(\mathrm{x}, \mathrm{y}\right) = 2{\left(\mathrm{x} - \mathrm{y}\right)}^2 - {\mathrm{x}}^4 - {\mathrm{y}}^4$ ${\left|\left({\mathrm{f}}_{\mathrm{xx}}{\mathrm{f}}_{\mathrm{yy}} - {\mathrm{f}}_{\mathrm{xy}}^2\right)\right|}_{\left(0, 0\right)}$

• 32

• 16

• 0

• - 1

If x is real, then the minimum value of $\frac{{\mathrm{x}}^2 - \mathrm{x} + 1}{{\mathrm{x}}^2 + \mathrm{x} + 1}$, is

• $\frac{1}{3}$

• 3

• $\frac{1}{2}$

• 1

A stone thrown upwards, has its equation of motion s = 490t - 4.9t². Then the maximum height reached by it, is

• 24500

• 12500

• 12250

• 25400