Stationary points

A curve has equation y = 3x⁴ - 4x³ - 6x² + 12x.

Use your answer to part (a)(i) to find the x-coordinates of the stationary points on the

curve.

Show that there is exactly one point on the curve which is a point of inflection but not a

stationary point.

The diagram shows the curve with equation y = (x³ - 2x²)ln x. The curve has a point of inflection at the point M.

Show that the x-coordinate of M satisfies the equation

Use an iterative formula, based on the equation in part (a)(i), to determine the x-coordinate of M correct to 2 decimal places. Use an initial value of 1.1 and show the

result of each step of the iterative process.

Determine the exact area of the shaded region, giving your answer in the form p ln q - r,

where p and r are positive rational numbers and q is a positive integer.

Determine the x-coordinates of any stationary points on the curve.

Show that the curve is convex for all values of x.

In this question you must show detailed reasoning.

A curve has equation y = x³ - 3x² + 4x.

Show that the curve has no stationary points.

Show that the curve has exactly one point of inflection.

The graph of the function y = f (x) passes through the point P with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to P on each of the following curves.

y = f (x) + 3

y = 2f (3 x - 1)

y = f^-1(x)

The diagram shows part of the graph of y = g ′ (x). This is the graph of the gradient function of y = g (x). The graph intersects the x-axis at x =- 2 and x = 4.

State the x-coordinate of any stationary points on the graph of y = g (x).

State the set of values of x for which y = g (x) is a decreasing function.

State the x-coordinate of any points of inflection on the graph of y = g (x).

The curve y = (x² - 2)ln x has one stationary point which is close to x = 1.

Show that the x-coordinate of this stationary point satisfies the equation 2x² ln x + x² - 2 = 0

Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a)

Apply the Newton-Raphson formula with initial value x₁ = 1 to find x₂ and x₃.

Find the coordinates of this stationary point, giving each coordinate correct to 3 decimal

places.

A curve has equation y = 2 ln(k - 3x) + x² - 3x, where k is a positive constant.

Given that the curve has a point of inflection where x = 1, show that k = 6.

It is also given that the curve intersects the x-axis at exactly one point.

Show by calculation that the x-coordinate of this point lies between 0.5 and 1.5.

Use the Newton-Raphson method, with initial value x₀ = 1, to find the x-coordinate of the point where the curve intersects the x-axis, giving your answer correct to 5 decimal places.

Show the result of each iteration to 6 decimal places.

By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.

The diagram shows the curve C with parametric equations

The maximum point on C is denoted by P.

Determine the exact coordinates of P.

The shaded region R is enclosed by the curve, the x-axis and the lines x = 1 and x = 6.

Show that the area of R is given by

where a and b are constants to be determined.

Hence determine the exact area of R.

The diagram shows the part of the curve y = 3x sin 2x for which 0 ⩽ x ⩽ .

The maximum point on the curve is denoted by P.

Show that the x-coordinate of P satisfies the equation tan 2 x + 2 x = 0.

Use the Newton-Raphson method, with a suitable initial value, to find the x-coordinate of P,

giving your answer correct to 4 decimal places. Show the result of each iteration.

The trapezium rule, with four strips of equal width, is used to find an approximation to

Explain briefly why it is not easy to tell from the diagram alone whether the trapezium rule with four strips of equal width gives an under-or over-estimate for the area of the

region in this case.

In this question you must show detailed reasoning.

Find the x-coordinate of the point where the curve crosses the x axis.

The points A and B lie on the curve and have x coordinates 2 and 4. Show that the line AB is parallel to

the x-axis.

Find the coordinates of the turning point on the curve.

Determine whether this turning point is a maximum or a minimum.