Find stationary points

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) on the curve \(C\) is a minimum turning point.
   Given that the \(x\) coordinate of \(P\) is 4
2. show that \(k = - 78\) The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\).
   The finite region \(R\), shown shaded in Figure 5, is bounded by \(C\), the \(y\)-axis and \(P N\).
3. Use integration to find the area of \(R\).

1. Given

$$y = 3 \sqrt { x } - 6 x + 4 , \quad x > 0$$

1. find \(\int y \mathrm {~d} x\), simplifying each term.
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. Hence find the value of \(x\) such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

5 In this question you must show detailed reasoning. A curve has equation \(y = x ^ { 3 } - 3 x ^ { 2 } + 4 x\).

1. Show that the curve has no stationary points.
2. Show that the curve has exactly one point of inflection.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) giving each term in simplest form.
   2. Hence determine the nature of the stationary point of \(C\), giving a reason for your answer.
4. State the range of values of \(x\) for which \(y\) is decreasing.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

7 It is given that \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { - x } ( 2 - x )\).
2. Hence find the exact coordinates of the stationary points on the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

9 A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x }\). Find the exact coordinates of the stationary points of the curve.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

A function is defined by \(f ( x ) = 2 x ^ { 3 } + 3 x - 5\). a) Prove that the graph of \(f ( x )\) does not have a stationary point.
b) Show that the graph of \(f ( x )\) does have a point of inflection and find the coordinates of the point of inflection.
c) Sketch the graph of \(f ( x )\). Evaluate the integral \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x\).

The curve \(y = x^2 - \frac{a}{x}\) has a stationary point at \((-3, b)\). Find the values of the constants \(a\) and \(b\). [4]

The equation of a curve is \(y = x^3 + px^2\), where \(p\) is a positive constant.

1. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of \(p\). [4]
2. Find the nature of each of the stationary points. [3]

Another curve has equation \(y = x^3 + px^2 + px\).

1. Find the set of values of \(p\) for which this curve has no stationary points. [3]

\includegraphics{figure_4} The diagram shows the curve \(y = e^x + 4e^{-2x}\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\). [3]
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0\), \(x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac{5}{2}\). [4]

\includegraphics{figure_6} The diagram shows the curve \(y = (4 - x)e^x\) and its maximum point \(M\). The curve cuts the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Write down the coordinates of \(A\) and \(B\). [2]
2. Find the \(x\)-coordinate of \(M\). [4]
3. The point \(P\) on the curve has \(x\)-coordinate \(p\). The tangent to the curve at \(P\) passes through the origin \(O\). Calculate the value of \(p\). [5]

A curve has equation \(y = 2\sin 2x - 5\cos 2x + 6\) and is defined for \(0 \leq x \leq \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. [6]

Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]

\includegraphics{figure_5} The diagram shows the curve \(y = e^{-x} - e^{-2x}\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(p\).

1. Find the exact value of \(p\). [4]
2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = p\) is equal to \(\frac{1}{4}\). [4]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A curve has equation $$y = 256x^4 - 304x - 35 + \frac{27}{x^2} \quad x \neq 0$$

1. Find \(\frac{dy}{dx}\) [3]
2. Hence find the coordinates of the stationary points of the curve. [5]

The diagram shows part of a curve with a maximum point \(M\). \includegraphics{figure_5} The equation of the curve is $$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$

1. Find \(\frac{dy}{dx}\). [3]
2. Hence find the coordinates of the maximum point \(M\). [4]
3. The point \(P(1, 14)\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20x - 6\). [3]
4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(RM\). [3]

Fig. 9 shows a sketch of the curve \(y = x^3 - 3x^2 - 22x + 24\) and the line \(y = 6x + 24\). \includegraphics{figure_9}

1. Differentiate \(y = x^3 - 3x^2 - 22x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places. [4]
2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = -4\). Find algebraically the \(x\)-coordinate of the other point of intersection. [3]
3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6x + 24\) for \(-4 \leq x \leq 0\), shown shaded on Fig. 9. [4]

Find the coordinates of the stationary point of the curve with equation $$y = x + \frac{4}{x^2}.$$ [6]

A curve has the equation $$y = x^3 + ax^2 - 15x + b,$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \((-1, 12)\),

1. find the values of \(a\) and \(b\), [6]
2. find the coordinates of the other stationary point of the curve. [3]

The curve \(C\) has the equation $$y = 3 - x^{\frac{1}{2}} - 2x^{-\frac{1}{2}}, \quad x > 0.$$

1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis. [4]
2. Find the exact coordinates of the stationary point of \(C\). [5]
3. Determine the nature of the stationary point. [2]
4. Sketch the curve \(C\). [2]