Find stationary points and nature

3. The curve \(C\) has equation $$y = 8 \sqrt { x } + \frac { 18 } { \sqrt { x } } - 20 \quad x > 0$$ a. Find
i) \(\frac { d y } { d x }\)
ii) \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\)
b. Use calculus to find the coordinates of the stationary point of \(C\).
c. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.

1. A curve \(C\) has equation

$$y = x ^ { 2 } - 2 x - 24 \sqrt { x } , \quad x > 0$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 4\)
3. Determine the nature of this stationary point, giving a reason for your answer.

10

1. A curve has equation \(y = 16 x + \frac { 1 } { x ^ { 2 } }\). Find
   (A) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
   (B) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Hence
   • find the coordinates of the stationary point,
   • determine the nature of the stationary point.

7 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 4\).
      (1 mark)
   2. Write \(\frac { 16 } { x ^ { 2 } }\) in the form \(16 x ^ { k }\), where \(k\) is an integer.
   3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   4. Hence determine whether the point where \(x = 4\) is a maximum or a minimum, giving a reason for your answer.
2. The point \(P ( 1,8 )\) lies on the curve.
   1. Show that the gradient of the curve at the point \(P\) is 12 .
   2. Find an equation of the normal to the curve at \(P\).
3. 1. Find \(\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x\).
   2. Hence find the equation of the curve which passes through the point \(P ( 1,8 )\).

2 The surface \(S\) has equation \(z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }\) for \(y \neq 0\).

1. (a) Find the following.
   • \(\frac { \partial z } { \partial x }\)
   • \(\frac { \partial z } { \partial y }\)
     (b) Find the coordinates of all stationary points of \(S\).
   • Find all four second partial derivatives of \(z\) with respect to \(x\) and/or \(y\).

6 A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\) and is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-4_545_784_420_628}

1. 1. Given that \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
   3. Find an equation of the normal to \(C\) at the point ( 1,6 ).
2. 1. Find \(\int \left( x + 1 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
   2. Hence find the area of the region bounded by the curve \(C\), the lines \(x = 1\) and \(x = 4\) and the \(x\)-axis.

8 The function g is defined by $$\mathrm { g } ( x ) = \mathrm { e } ^ { \sin x } \quad ( 0 \leq x \leq 2 \pi )$$ The diagram below shows the graph of \(y = \mathrm { g } ( x )\)
\includegraphics[max width=\textwidth, alt={}, center]{a9f88195-e545-43f2-a13a-6459d14e1cda-09_369_593_548_721} 8

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = \mathrm { g } ( x )\), giving your answers in exact form. 8
2. Use Simpson's rule with 3 ordinates to estimate $$\int _ { 0 } ^ { \pi } g ( x ) d x$$ giving your answer to two decimal places.
   8
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b).