Application of Derivatives
58 Questions Available
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Show that(-∞, 1], x > - 1, is an increasing function of x throughout its domain.
Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.
Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
Find the intervals in which the following function is (a) increasing (b) decreasing:
Prove that the area of a right angled triangle of given hypotenuse is maximum when the triangle is isosceles.
Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.
Let I be any interval disjoint from (- 1, 1). Prove that the function f given by f(x) = x + 1/x is strictly increasing on I.
Find the intervals in which the function f given by f (x) = 2x3 – 3x2 – 36x + 7 is (a) strictly increasing (b) strictly decreasing.
Find the equation of the tangent to the curve
The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?
A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan –1(0.5). Water is poured into it at a constant rate of 5 cubic meter per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m.
The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?
Find the intervals in which the function f given by [2, ∞), xy = (x² - 1) (x – 2)0 is (i) increasing (ii) decreasing.
Find intervals in which the function given by [8/5, 2] is (a) strictly increasing (b) strictly decreasing.
A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post which is 6 metres high. Find the rate at which the length of his shadow increases.
A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y - coordinate is changing 8 times as fast as the x-coordinate.
Find the intervals in which the following function is (a) increasing (b) decreasing:
