Stationary points

y 4x³ 5x² 2x

Find the coordinates of the turning points on the graph with

18A curve C has equation y = x³ ‒ 40x + 1

Find the coordinates of both the points on C at which the gradient is 8

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.

16A curve has equation y 4x³ 8x 5 Find the x coordinates of the two points on the curve where the gradient is

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.

Express 2x² – 12x + 3 in the form a(x + b)² + cwhere a, b and c are integers.

The curve C has equation y = 2(x + 4)² – 12(x + 4) + 3

The point M is the minimum point on C

Find the coordinates of M

Express 7 + 12x – 3x² in the form a + b(x + c)² where a, b and c are integers.

C is the curve with equation y = 7 + 12x – 3x² The point A is the maximum point on C

Use your answer to part (a) to write down the coordinates of A

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.

23Curve C has equation y = px³ − mx where p and m are positive integers.

Find the range of values of x, in terms of p and m, for which the gradient of C is negative.

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.

The curve shown in the diagram has equation

y = x³x + k where k is a positive constant with k 54

The curve has a maximum point at A (a, b) The curve has a minimum point at B (c, d)

Using differentiation, find the value of bd Show your working clearly.

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.