Differentiating powers of x

A curve C has equation y x³ 8x² 12x 5

Curve C has exactly two stationary points, one at point A and one at point B such that

x coordinate of point A > x coordinate of point B

Find the coordinates of point A Show clear algebraic working.

The curve C has equation y = 4x³ + x² – 20x

Find the x coordinates of the points on C where the gradient is 4 Show clear algebraic working.

21The curve T has equation y = x³ – 2x² – 9x + 15

Find the range of values of x for which T has a positive gradient. Give your values correct to 3 significant figures. Show your working clearly.

The radius of a right circular cylinder is x cm.

The volume of the cylinder is Vcm³

Find the maximum value of V Give your answer correct to the nearest whole number.

G is the point on the curve with equation y = 8x² – 14x – 6 where the gradient is 10 The straight line Q passes through the point G and is perpendicular to the tangent at G

Find an equation for Q Give your answer in the form ax + by + c = 0 where a, b and c are integers.

A particle P moves along a straight line. The fixed point O lies on this line.

The displacement of P from O at time t seconds, t 1, is s metres where

The velocity of P at time t seconds, t 1, is vm / s Work out the distance of P from O at the instant when v = 0

A solid, S, is made from a hemisphere and a cylinder.

The centre of the circular face of the hemisphere and the centre of the top face of the cylinder are at the same point.

The radius of the cylinder and the radius of the hemisphere are both xcm. The height of the cylinder is (20 – 4x)cm.

Find the maximum value of y. Show clear algebraic working.

A particle P is moving along a straight line. The fixed point O lies on the line.

At time t seconds (t 0), the displacement of P from O is s metres where

Find the minimum speed of P.

There are two points on the curve C at which the gradient of the curve is 2

Find the x coordinate of each of these two points. Show clear algebraic working.

The diagram shows a cuboid of volume Vcm³

Show that V= 15 + 16x−x²− 2x³

There is a value of x for which the volume of the cuboid is a maximum.

Find this value of x. Show your working clearly. Give your answer correct to 3 significant figures.

A particle P is moving along a straight line. The fixed point O lies on this line.

At time t seconds, the displacement, s metres, of P from O is given by

At time t seconds, the velocity of P is v m/s.

Find an expression for v in terms of t.

Find the time at which the acceleration of the particle is 6 m/s²

y = x³ – 2x² – 15x + 5

C is the curve with equation y = x³ – 2x² – 15x + 5

Work out the range of values of x for which C has a negative gradient.