Find curve from second derivative

11 The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(f ^ { \prime \prime } ( x ) = ( 4 x + 1 ) ^ { - \frac { 1 } { 2 } }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
   It is now given that \(f ^ { \prime \prime } ( 0 ) , f ^ { \prime } ( 0 )\) and \(f ( 0 )\) are the first three terms respectively of an arithmetic progression.
2. Find the value of \(\mathrm { f } ( 0 )\).
3. Find \(\mathrm { f } ( x )\), and hence find the minimum value of f .

10 A curve for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5\) has a stationary point at \(( 3,6 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the other stationary point on the curve.
3. Determine the nature of each of the stationary points.
   \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve \(y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }\) and a point \(P ( 2,1 )\) lying on the curve. The normal to the curve at \(P\) intersects the \(x\)-axis at \(Q\).
4. Show that the \(x\)-coordinate of \(Q\) is \(\frac { 16 } { 9 }\).
5. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11. A curve has equation \(y = \mathrm { f } ( x )\), where $$f ^ { \prime \prime } ( x ) = \frac { 6 } { \sqrt { x ^ { 3 } } } + x \quad x > 0$$ The point \(P ( 4 , - 50 )\) lies on the curve.
Given that \(\mathrm { f } ^ { \prime } ( x ) = - 4\) at \(P\),

1. find the equation of the normal at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
   (3)
2. find \(\mathrm { f } ( x )\).
   (8)

   VIIIV SIHI NI IIIYM ION OC

   VIIV SIHI NI JIHMM ION OO

   VI4V SIHI NI JIIYM ION OO

1. A curve has equation \(y = \mathrm { f } ( x )\).

The point \(P \left( 4 , \frac { 32 } { 3 } \right)\) lies on the curve.
Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3\)
• \(\quad \mathrm { f } ^ { \prime } ( x ) = 5\) at \(P\)
  find
  1. the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found,
  2. \(\mathrm { f } ( x )\).
     

5 A curve has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }\) for all positive values of \(x\). The point \(P ( 4,1 )\) lies on the curve, and the gradient of the curve at \(P\) is 5 . Find the equation of the curve.