Determine increasing/decreasing intervals

11 It is given that a curve has equation \(y = k ( 3 x - k ) ^ { - 1 } + 3 x\), where \(k\) is a constant.

1. Find, in terms of \(k\), the values of \(x\) at which there is a stationary point.
   The function f has a stationary value at \(x = a\) and is defined by $$f ( x ) = 4 ( 3 x - 4 ) ^ { - 1 } + 3 x \quad \text { for } x \geqslant \frac { 3 } { 2 }$$
2. Find the value of \(a\) and determine the nature of the stationary value.
3. The function g is defined by \(\mathrm { g } ( x ) = - ( 3 x + 1 ) ^ { - 1 } + 3 x\) for \(x \geqslant 0\). Determine, making your reasoning clear, whether \(g\) is an increasing function, a decreasing function or neither.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 The function f is defined by \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. State, with a reason, whether f is an increasing function, a decreasing function or neither. The function g is defined by \(\mathrm { g } ( x ) = \frac { 1 } { x + 1 } + \frac { 1 } { ( x + 1 ) ^ { 2 } }\) for \(x < - 1\).
3. Find the coordinates of the stationary point on the curve \(y = \mathrm { g } ( x )\).

4. (i) $$f ( x ) = \frac { ( 2 x + 5 ) ^ { 2 } } { x - 3 } \quad x \neq 3$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\) in the form \(\frac { P ( x ) } { Q ( x ) }\) where \(P ( x )\) and \(Q ( x )\) are fully factorised quadratic expressions.
2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
   (ii) $$g ( x ) = x \sqrt { \sin 4 x } \quad 0 \leqslant x < \frac { \pi } { 4 }$$ The curve with equation \(y = g ( x )\) has a maximum at the point \(M\). Show that the \(x\) coordinate of \(M\) satisfies the equation $$\tan 4 x + k x = 0$$ where \(k\) is a constant to be found.
   

1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) > 0\) You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

2.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), writing your answer as a single fraction in its simplest form.
2. Hence find the set of values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\) 2. $$y = \frac { 4 x } { x ^ { 2 } + 5 }$$

7. $$\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$

1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
3. Use Simpson's rule with six strips to find an approximate value for $$\int _ { 0 } ^ { 6 } f ( x ) d x$$

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

1. The function \(g\) is defined by

$$g ( x ) = \frac { 3 \ln ( x ) - 7 } { \ln ( x ) - 2 } \quad x > 0 \quad x \neq k$$ where \(k\) is a constant.

1. Deduce the value of \(k\).
2. Prove that $$\mathrm { g } ^ { \prime } ( x ) > 0$$ for all values of \(x\) in the domain of g .
3. Find the range of values of \(a\) for which $$g ( a ) > 0$$

3. $$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$

1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

10 Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }\) is an increasing function for all values of \(x\).