Determine if function is increasing/decreasing

2 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { 3 } ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 2 x\) for \(\frac { 1 } { 2 } < x < a\). It is given that f is a decreasing function. Find the maximum possible value of the constant \(a\).

9 The equation of a curve is \(y = 4 + 5 x + 6 x ^ { 2 } - 3 x ^ { 3 }\).

1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases.
2. It is given that \(y = 9 x + k\) is a tangent to the curve. Find the value of the constant \(k\).

10 The equation of a curve is \(y = \frac { 1 } { 6 } ( 2 x - 3 ) ^ { 3 } - 4 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis.
3. Find the set of values of \(x\) for which \(\frac { 1 } { 6 } ( 2 x - 3 ) ^ { 3 } - 4 x\) is an increasing function of \(x\).

10 It is given that a curve has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x\).

1. Find the set of values of \(x\) for which the gradient of the curve is less than 5 .
2. Find the values of \(\mathrm { f } ( x )\) at the two stationary points on the curve and determine the nature of each stationary point.

10 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 5 x\) passes through the origin.

1. Show that the curve has no stationary points.
2. Denoting the gradient of the curve by \(m\), find the stationary value of \(m\) and determine its nature.
3. Showing all necessary working, find the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = 6\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A function f is defined by \(\mathrm { f } : x \mapsto x ^ { 3 } - x ^ { 2 } - 8 x + 5\) for \(x < a\). It is given that f is an increasing function. Find the largest possible value of the constant \(a\).

4 The function f is such that \(\mathrm { f } ( x ) = ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 6 x\) for \(\frac { 1 } { 2 } < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which f is a decreasing function.

2 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7\) for \(x \geqslant - 2\). Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.

2 An increasing function, f , is defined for \(x > n\), where \(n\) is an integer. It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 6 x + 8\). Find the least possible value of \(n\).

9. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geqslant 0$$

1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
   The line \(l\) is the tangent to \(C\) at the point \(P\).
   The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
2. Find, using calculus, the exact area of \(R\).

1. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 8 x + 5$$

1. Find \(f ^ { \prime \prime } ( x )\)
2. 1. Solve \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\)
   2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is concave.

9 A function f is defined for \(x > 0\) by \(\mathrm { f } ( x ) = \frac { 6 } { x ^ { 2 } + a }\), where \(a\) is a positive constant.

1. Show that f is a decreasing function.
2. Find, in terms of \(a\), the coordinates of the point of inflection on the curve \(y = \mathrm { f } ( x )\).