Gas is escaping from a spherical balloon at the rate of 900 cm³/sec. How fast is the surface area and radius of the balloon shrinking when the radius of the balloon is 30 cm?

A particle moves along the curve \(y=\frac { 4 }{ 3 } { x }^{ 3 }+5\). Find the points on the curve at which y coordinate changes as fast as x-coordinates.

Find the points on the curve y=2x²-2x² at which the tangent lines are parallel to the line y=3x-2.

If the curves 4x=y² and 4xy=k cut at right angles show that k²=512.

missle fired from ground level rises x metres vertically upwards in t seconds and \(x=100t-\frac { 25 }{ 2 } { t }^{ 2 }\). Find the 
(i) initial velocity of the missile
(ii) the time when the height of the missile is maximum
(iii) the maximum height reached
(iv) the velocity which the missile strikes the ground.

A water tank has a shape of an inverted cone with its axis vertical and vertex lower most. Its semi vertical angle is tan⁻¹(0.5). Water is poured into it at a constant rate of 5 cm3/hr. Find the rate at which the level of the water is rising at that instant when the depth of the water is 4 m.

Evaluate:\(\underset { x\rightarrow 1 }{ lim } \frac { 1 }{ { x }^{ x-1 } } \)

If f(x)=alogx+bx²+x has exterme at x=1, x=2 then find a and b.

Find the intervals of concavity and points of inflexion for f(x)=x³-15x²+75x-50.

Find the intervals for which the function f(x)=2x²-9x²-12x+1 is increasing or decfreasing and find the local extermems.

Find the local maximum and local minimum values for f(x)=12x²-2x²-x⁴.

Find the intervals of concavity and the points of inflection of f(x)=12x²-2x³-x⁴.

Find the points of inflection and determine the intervals of concavity of \(y={ e }^{ -{ x }^{ 2 } }\)

A manufacturer can sell x items at a price of rupees \(\left( 5-\frac { x }{ 100 } \right) \) each. The cost price of x items is Rs.\(\left( \frac { x }{ 5 } +500 \right) \) .Find the numbers of items he should sell to earn maximum profit.

Find the local maximum and local minimum values of f(x)=x⁴-3x+3x²-x.