Increasing/decreasing intervals

1 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 x + 2 } + x ^ { 2 }\) for \(x < - 1\).
Determine whether f is an increasing function, a decreasing function or neither.

1 It is given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ^ { 3 } + x\), for \(x \in \mathbb { R }\). Show that f is an increasing function.

2 A curve has equation \(y = 3 x ^ { 3 } - 6 x ^ { 2 } + 4 x + 2\). Show that the gradient of the curve is never negative.

2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.

Given \(y = \frac { x ^ { 3 } } { 3 } - 2 x ^ { 2 } + 3 x + 5\)

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), simplifying each term.
2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 0\)

3. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).

7 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\).

2 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.

9 Use calculus to find the set of values of \(x\) for which \(\mathrm { f } ( x ) = 12 x - x ^ { 3 }\) is an increasing function.

4 Differentiate \(2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\). Hence find the set of values of \(x\) for which the function \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\) is increasing.

5 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.

10 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x\). Find the set of values of \(x\) for which \(y\) is an increasing function of \(x\).

3 Find the set of values of \(x\) for which \(x ^ { 2 } - 7 x\) is a decreasing function.

6 Differentiate \(2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\). Hence find the set of values of \(x\) for which the function \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 24 x\) is increasing.

5 Use calculus to find the set of values of \(x\) for which \(x ^ { 3 } - 6 x\) is an increasing function.

11 Chris claims that, "for any given value of \(x\), the gradient of the curve
\(y = 2 x ^ { 3 } + 6 x ^ { 2 } - 12 x + 3\) is always greater than the gradient of the curve
\(y = 1 + 60 x - 6 x ^ { 2 \prime \prime }\).
Show that Chris is wrong by finding all the values of \(x\) for which his claim is not true.
[0pt] [7 marks]

1. A curve has equation

$$y = 3 x ^ { 2 } + \frac { 24 } { x } + 2 \quad x > 0$$

1. Find, in simplest form, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the exact range of values of \(x\) for which the curve is increasing.

1. A curve has equation

$$y = \frac { 2 } { 3 } x ^ { 3 } - \frac { 7 } { 2 } x ^ { 2 } - 4 x + 5$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) writing your answer in simplest form.
2. Hence find the range of values of \(x\) for which \(y\) is decreasing.

1. A curve has equation

$$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.

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3 The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a quadratic function of \(x\).
A copy of the diagram is given in the Printed Answer Booklet.
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1. On the copy of the diagram in the Printed Answer Booklet, draw a possible graph of the gradient function \(y = \mathrm { f } ^ { \prime } ( x )\).
2. State the gradient of the graph of \(y = \mathrm { f } ^ { \prime \prime } ( x )\).

10 In this question you must show detailed reasoning.

1. Sketch the gradient function for the curve \(y = 24 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Determine the set of values of \(x\) for which \(24 x - 3 x ^ { 2 } - x ^ { 3 }\) is decreasing.

13 Determine the range of values of \(x\) for which \(y = 4 x ^ { 3 } + 7 x ^ { 2 } - 6 x + 8\) is a decreasing function.

7 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve \(C\) at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 20 x - 6 x ^ { 2 } - 16$$

1. 1. Show that \(y\) is increasing when \(3 x ^ { 2 } - 10 x + 8 < 0\).
   2. Solve the inequality \(3 x ^ { 2 } - 10 x + 8 < 0\).
2. The curve \(C\) passes through the point \(P ( 2,3 )\).
   1. Verify that the tangent to the curve at \(P\) is parallel to the \(x\)-axis.
   2. The point \(Q ( 3 , - 1 )\) also lies on the curve. The normal to the curve at \(Q\) and the tangent to the curve at \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
      (7 marks)

2. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.