1.07n Stationary points: find maxima, minima using derivatives

11 \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-18_387_920_260_609} The diagram shows part of the curve with equation \(y = x ^ { 3 } - 2 b x ^ { 2 } + b ^ { 2 } x\) and the line \(O A\), where \(A\) is the maximum point on the curve. The \(x\)-coordinate of \(A\) is \(a\) and the curve has a minimum point at ( \(b , 0\) ), where \(a\) and \(b\) are positive constants.

1. Show that \(b = 3 a\).
2. Show that the area of the shaded region between the line and the curve is \(k a ^ { 4 }\), where \(k\) is a fraction to be found.
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11 The equation of a curve is \(y = 2 \sqrt { 3 x + 4 } - x\).

1. Find the equation of the normal to the curve at the point (4,4), giving your answer in the form \(y = m x + c\).
2. Find the coordinates of the stationary point.
3. Determine the nature of the stationary point.
4. Find the exact area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\).
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11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).

1. Find the value of \(k\).
   ................................................................................................................................................. . .
2. Find the equation of the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).

11 \includegraphics[max width=\textwidth, alt={}, center]{aaba3158-b5be-464e-bea3-1a4c460f9637-16_622_1091_260_525} The diagram shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } + k ^ { 2 } x ^ { - \frac { 1 } { 2 } }\), where \(k\) is a positive constant.

1. Find the coordinates of the minimum point of the curve, giving your answer in terms of \(k\).
   The tangent at the point on the curve where \(x = 4 k ^ { 2 }\) intersects the \(y\)-axis at \(P\).
2. Find the \(y\)-coordinate of \(P\) in terms of \(k\).
   The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = \frac { 9 } { 4 } k ^ { 2 }\) and \(x = 4 k ^ { 2 }\).
3. Find the area of the shaded region in terms of \(k\).
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10 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x ^ { 2 } - \frac { 4 } { x ^ { 3 } }\). The curve has a stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).

1. Determine the nature of the stationary point at \(\left( - 1 , \frac { 9 } { 2 } \right)\).
2. Find the equation of the curve.
3. Show that the curve has no other stationary points.
4. A point \(A\) is moving along the curve and the \(y\)-coordinate of \(A\) is increasing at a rate of 5 units per second. Find the rate of increase of the \(x\)-coordinate of \(A\) at the point where \(x = 1\).
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9 The equation of a curve is \(y = 3 x + 1 - 4 ( 3 x + 1 ) ^ { \frac { 1 } { 2 } }\) for \(x > - \frac { 1 } { 3 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point of the curve and determine its nature.

11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).

1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
   The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
2. Find the coordinates of \(Q\) in terms of \(m\).
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11 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - 30 x + 6 a\), where \(a\) is a positive constant. The curve has a stationary point at \(( a , - 15 )\).

1. Find the value of \(a\).
2. Determine the nature of this stationary point.
3. Find the equation of the curve.
4. Find the coordinates of any other stationary points on the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-14_832_830_276_653} The diagram shows the points \(A \left( 1 \frac { 1 } { 2 } , 5 \frac { 1 } { 2 } \right)\) and \(B \left( 7 \frac { 1 } { 2 } , 3 \frac { 1 } { 2 } \right)\) lying on the curve with equation \(y = 9 x - ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find the coordinates of the maximum point of the curve.
2. Verify that the line \(A B\) is the normal to the curve at \(A\).
3. Find the area of the shaded region.
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5 The equation of a curve is \(y = 2 x ^ { 2 } - \frac { 1 } { 2 x } + 3\).

1. Find the coordinates of the stationary point.
2. Determine the nature of the stationary point.
3. For positive values of \(x\), determine whether the curve shows a function that is increasing, decreasing or neither. Give a reason for your answer.

10 The gradient of a curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( x + 3 ) ^ { \frac { 1 } { 2 } } - x\). The curve has a stationary point at \(( a , 14 )\), where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Determine the nature of the stationary point.
3. Find the equation of the curve.

11 \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-18_497_1049_264_548} The diagram shows the curve with equation \(y = 9 \left( x ^ { - \frac { 1 } { 2 } } - 4 x ^ { - \frac { 3 } { 2 } } \right)\). The curve crosses the \(x\)-axis at the point \(A\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the equation of the tangent to the curve at \(A\).
3. Find the \(x\)-coordinate of the maximum point of the curve.
4. Find the area of the region bounded by the curve, the \(x\)-axis and the line \(x = 9\).
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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.
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10 At the point \(( 4 , - 1 )\) on a curve, the gradient of the curve is \(- \frac { 3 } { 2 }\). It is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { - \frac { 1 } { 2 } } + k\), where \(k\) is a constant.

1. Show that \(k = - 2\).
2. Find the equation of the curve.
3. Find the coordinates of the stationary point.
4. Determine the nature of the stationary point.

11 \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-14_467_757_262_653} The diagram shows the curve with equation \(\mathrm { y } = 2 \mathrm { x } ^ { - \frac { 2 } { 3 } } - 3 \mathrm { x } ^ { - \frac { 1 } { 3 } } + 1\) for \(x > 0\). The curve crosses the \(x\)-axis at points \(A\) and \(B\) and has a minimum point \(M\).

1. Find the exact coordinates of \(M\).
2. Find the area of the region bounded by the curve and the line segment \(A B\).
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12 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-18_557_677_264_733} The diagram shows a curve with equation \(y = 4 x ^ { \frac { 1 } { 2 } } - 2 x\) for \(x \geqslant 0\), and a straight line with equation \(y = 3 - x\). The curve crosses the \(x\)-axis at \(A ( 4,0 )\) and crosses the straight line at \(B\) and \(C\).

1. Find, by calculation, the \(x\)-coordinates of \(B\) and \(C\).
2. Show that \(B\) is a stationary point on the curve.
3. Find the area of the shaded region.
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10 \includegraphics[max width=\textwidth, alt={}, center]{6bcc553c-4938-46ef-bba4-97391b4d58d4-14_378_666_264_737} The diagram shows part of the curve \(y = \frac { 2 } { ( 3 - 2 x ) ^ { 2 } } - x\) and its minimum point \(M\), which lies on the \(x\)-axis.

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y \mathrm {~d} x\).
2. Find, by calculation, the \(x\)-coordinate of \(M\).
3. Find the area of the shaded region bounded by the curve and the coordinate axes.

8 The equation of a curve is \(y = 2 x + 1 + \frac { 1 } { 2 x + 1 }\) for \(x > - \frac { 1 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point and determine the nature of the stationary point.

9 A curve has equation \(y = \mathrm { f } ( x )\), and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } - 7 - \frac { 4 } { x ^ { 2 } }\).

1. Given that \(\mathrm { f } ( 1 ) = - \frac { 1 } { 3 }\), find \(\mathrm { f } ( x )\).
2. Find the coordinates of the stationary points on the curve.
3. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
4. Hence, or otherwise, determine the nature of each of the stationary points.

10 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } + \frac { k } { x } + 2\) for \(x > 0\).

1. Given that the curve with equation \(y = \mathrm { f } ( x )\) has a stationary point when \(x = 2\), find \(k\).
2. Determine the nature of the stationary point.
3. Given that this is the only stationary point of the curve, find the range of f .

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
   It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
2. Find \(\mathrm { f } ( x )\).
3. Find the coordinates of the other stationary point and determine its nature.
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3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).

8 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(( 3,5 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point.
3. State the set of values of \(x\) for which \(y\) increases as \(x\) increases.

10 A curve has a stationary point at \(( 2 , - 10 )\) and is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 6 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the curve.
3. Find the coordinates of the other stationary point and determine its nature.
4. Find the equation of the tangent to the curve at the point where the curve crosses the \(y\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-18_689_828_276_646} The diagram shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 40\). Parallel tangents, each with gradient 1 , touch the circle at points \(A\) and \(B\).
   1. Find the equation of the line \(A B\), giving the answer in the form \(y = m x + c\).
   2. Find the coordinates of \(A\), giving each coordinate in surd form.
   3. Find the equation of the tangent at \(A\), giving the answer in the form \(y = m x + c\), where \(c\) is in surd form.
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11 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-14_693_782_267_669} The diagram shows part of the curve with equation \(y = x + \frac { 2 } { ( 2 x - 1 ) ^ { 2 } }\). The lines \(x = 1\) and \(x = 2\) intersect the curve at \(P\) and \(Q\) respectively and \(R\) is the stationary point on the curve.

1. Verify that the \(x\)-coordinate of \(R\) is \(\frac { 3 } { 2 }\) and find the \(y\)-coordinate of \(R\).
2. Find the exact value of 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.