Classify nature of stationary points

10

1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
3. Determine whether each stationary point is a maximum point or a minimum point.
4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).

10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve \(\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }\) has a stationary point near \(x = 3\).

• Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
• Show that this stationary point is a maximum turning point.

5 In this question you must show detailed reasoning.

1. Find the coordinates of the two stationary points on the graph of \(y = 15 - x ^ { 2 } - \frac { 16 } { x ^ { 2 } }\).
2. Show that both these stationary points are maximum points.

2 The curve with equation \(y = x ^ { 4 } - 32 x + 5\) has a single stationary point, \(M\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(x\)-coordinate of \(M\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence, or otherwise, determine whether \(M\) is a maximum or a minimum point.
4. Determine whether the curve is increasing or decreasing at the point on the curve where \(x = 0\).

3 Two numbers, \(x\) and \(y\), are such that \(3 x + y = 9\), where \(x \geqslant 0\) and \(y \geqslant 0\). It is given that \(V = x y ^ { 2 }\).

1. Show that \(V = 81 x - 54 x ^ { 2 } + 9 x ^ { 3 }\).
2. 1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} x } = k \left( x ^ { 2 } - 4 x + 3 \right)\), and state the value of the integer \(k\).
   2. Hence find the two values of \(x\) for which \(\frac { \mathrm { d } V } { \mathrm {~d} x } = 0\).
3. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
4. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\) for each of the two values of \(x\) found in part (b)(ii).
   2. Hence determine the value of \(x\) for which \(V\) has a maximum value.
   3. Find the maximum value of \(V\).

3 The curve with equation \(y = x ^ { 5 } + 20 x ^ { 2 } - 8\) passes through the point \(P\), where \(x = - 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Verify that the point \(P\) is a stationary point of the curve.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
4. Find an equation of the tangent to the curve at the point where \(x = 1\).

9

1. Find the exact coordinates of the turning points of \(C\).
   Determine the nature of each turning point.
   Fully justify your answer.
   9
2. State the coordinates of the turning points of the curve $$y = \mathrm { f } ( x + 1 ) - 4$$

2 A curve has equation \(y = \mathrm { f } ( x )\) The curve has a point of inflection at \(x = 7\) It is given that \(\mathrm { f } ^ { \prime } ( 7 ) = a\) and \(\mathrm { f } ^ { \prime \prime } ( 7 ) = b\), where \(a\) and \(b\) are real numbers. Identify which one of the statements below must be true.
Circle your answer. \(\mathrm { f } ^ { \prime } ( 7 ) \neq 0\) \(\mathrm { f } ^ { \prime } ( 7 ) = 0\) \(\mathrm { f } ^ { \prime \prime } ( 7 ) \neq 0\) \(\mathrm { f } ^ { \prime \prime } ( 7 ) = 0\)

The equation of a curve is \(y = f(x)\), where \(f(x) = (4x - 3)^{\frac{3}{4}} - \frac{20}{3}x\).

1. Find the \(x\)-coordinates of the stationary points of the curve and determine their nature. [6]
2. State the set of values for which the function f is increasing. [1]

A curve has equation \(y = (2x - 1)^{-1} + 2x\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]

A curve is such that \(\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4\). The curve has a stationary point at \(P\) where \(x = 2\).

1. State, with a reason, the nature of this stationary point. [1]
2. Find an expression for \(\frac{dy}{dx}\). [4]
3. Given that the curve passes through the point \((1, 13)\), find the coordinates of the stationary point \(P\). [4]

A curve \(y = f(x)\) has a stationary point at \((3, 7)\) and is such that \(f''(x) = 36x^{-3}\).

1. State, with a reason, whether this stationary point is a maximum or a minimum. [1]
2. Find \(f'(x)\) and \(f(x)\). [7]

The curve \(C\) has equation \(y = 2x^3 - 5x^2 - 4x + 2\).

1. Find \(\frac{dy}{dx}\). [2]
2. Using the result from part (a), find the coordinates of the turning points of \(C\). [4]
3. Find \(\frac{d^2y}{dx^2}\). [2]
4. Hence, or otherwise, determine the nature of the turning points of \(C\). [2]

A curve has equation \(y = 2x^2 - 8x\sqrt{x} + 8x + 1\) for \(x \geq 0\)

1. Prove that the curve has a maximum point at \((1, 3)\) Fully justify your answer. [9 marks]
2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]

A continuous curve has equation \(y = f(x)\) The curve passes through the points \(A(2, 1)\), \(B(4, 5)\) and \(C(6, 1)\) It is given that \(f'(4) = 0\) Jasmin made two statements about the nature of the curve \(y = f(x)\) at the point \(B\): Statement 1: There is a turning point at \(B\) Statement 2: There is a maximum point at \(B\)

1. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is correct. [1 mark]
2. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is not correct. [1 mark]
3. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is not correct and Statement 2 is not correct. [1 mark]

It is given that for the continuous function \(g\) • \(g'(1) = 0\) • \(g'(4) = 0\) • \(g''(x) = 2x - 5\)

1. Determine the nature of each of the turning points of \(g\) Fully justify your answer. [3 marks]
2. Find the set of values of \(x\) for which \(g\) is an increasing function. [2 marks]

The function f is defined by $$f(x) = x^2 + 2 \cos x \text{ for } -\pi \leq x \leq \pi$$ Determine whether the curve with equation \(y = f(x)\) has a point of inflection at the point where \(x = 0\) Fully justify your answer. [4 marks]

A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)

1. 1. Find \(f''(x)\) [2 marks]
   2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
   1. State the single transformation which maps f onto g. [1 mark]
   2. State the range of values of \(x\) for which g is an increasing function. [1 mark]

A curve \(C\) has equation \(f(x) = 3x^3 - 5x^2 + x - 6\).

1. Find the coordinates of the stationary points of \(C\) and determine their nature. [8]
2. Without solving the equations, determine the number of distinct real roots for each of the following:
   1. \(3x^3 - 5x^2 + x + 1 = 0\),
   2. \(6x^3 - 10x^2 + 2x + 1 = 0\). [4]

A curve C has equation \(y = -x^3 + 12x - 20\).

1. Find the coordinates of the stationary points of C and determine their nature. [7]
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation. [3]

A curve C has equation \(f(x) = 5x^3 + 2x^2 - 3x\).

1. Find the \(x\)-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary. [5]
2. Determine the range of values of \(x\) for which C is concave. [2]

A curve has equation \(y = x^5 - 5x^4\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Verify that the curve has a stationary point when \(x = 4\). [2]
3. Determine the nature of this stationary point. [2]