Second derivative and nature determination

7 The function f is defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   The first, second and third terms of a geometric progression are respectively \(\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )\) and \(k \mathrm { f } ^ { \prime \prime } ( 2 )\).
2. Find the value of the constant \(k\).

1. Given that

$$y = 5 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } - 7 x \quad x > 0$$ find, in simplest form,

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)

4. \(y = \arctan ( \sqrt { } x ) , \quad x > 0,0 < y < \frac { \pi } { 2 }\).

1. Find the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(\mathrm { x } = \frac { 1 } { 4 }\).
2. Show that \(2 x ( 1 + x ) \frac { d ^ { 2 } y } { d x ^ { 2 } } + ( 1 + 3 x ) \frac { d y } { d x } = 0\).

9 Given that \(\mathrm { f } ( x ) = \frac { 1 } { x } - \sqrt { x } + 3\),

1. find \(\mathrm { f } ^ { \prime } ( x )\),
2. find \(\mathrm { f } ^ { \prime \prime } ( 4 )\).

6 Given that \(\mathrm { f } ( x ) = \frac { 4 } { x } - 3 x + 2\),

1. find \(\mathrm { f } ^ { \prime } ( x )\),
2. find \(\mathrm { f } ^ { \prime \prime } \left( \frac { 1 } { 2 } \right)\).

1 Given that \(y = x ^ { 5 } + \frac { 1 } { x ^ { 2 } }\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

1. $$y = 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 10$$

1. Find in simplest form
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Hence find the exact value of \(x\) when \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\)

4 Find the second derivative of \(\left( x ^ { 2 } + 5 \right) ^ { 4 }\), giving your answer in factorised form.

3 A curve has equation \(y = k \sqrt { x }\) where \(k\) is a constant. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 4,2 k )\) on the curve, giving your answer as an expression in terms of \(k\).

A curve has equation $$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$ where \(a\), \(b\) and \(c\) are positive constants. The curve has a single turning point. Use the second derivative of \(y\) to determine the nature of this turning point. You do not need to find the coordinates of the turning point. Fully justify your answer. [7 marks]