The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is 

The function sin⁴ x + cos⁴ x is increasing in the interval

One of the closest points on the curve x² - y² = 4 to the point (6, 0) is

The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________

The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

Which of the following statement is incorrect?
(1) Initial velocity means velocity at t = 0.
(2) Initial acceleration means, acceleration at t = 0.
(3) If the motion is upward, at the maximum height the velocity is not zero.
(4) If the motion is horizontal, u = 0 when the particle comes to rest.

The temperature T in celsius in a long rod of length 10 m, insulated at both ends, is a function of length x given by T = x(10 − x). Prove that the rate of change of temperature at the midpoint of the rod is zero.

A particle moves so that the distance moved is according to the law s(t) = \(s(t)=\frac{t^{3}}{3}-t^{2}+3\). At what time the velocity and acceleration are zero.

Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).

A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.

Find the tangent and normal to the following curves at the given points on the curve
y = x sin x at \(\left( \frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right) \)

Find the absolute maximum and absolute minimum values of the function f (x) = 2x³ + 3x² −12x on [−3, 2]

The ends of a rod AB which is 5 m long moves along two grooves OX, OY which at the right angles. If A moves at a constant speed of \(\frac { 1 }{ 2 } \) m/sec, what is the speed of B, when it is 4m from O?

A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A 10 kilometres to the north of P and traveling at 80 km/hr, while car B is 15 kilometres to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing?

Find the angle between the curves y = x² and x = y² at their points of intersection (0,0) and (1,1).

Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan^-1(\(\sqrt { 2 } \)).