1.07n Stationary points: find maxima, minima using derivatives

9 A curve has equation $$y = \mathrm { e } ^ { 3 x } - 5 \mathrm { e } ^ { 2 x } + 8 \mathrm { e } ^ { x }$$

1. Find the exact coordinates of the stationary points of \(y\).
2. Determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } > 0$$
3. Determine the nature of the stationary points on the curve.

7 A curve has equation \(y = \frac { 4 x + 11 } { ( x + 3 ) ^ { 2 } }\).

1. Show that the curve meets the line \(y = k\) if and only if \(k \leq 4\), and deduce the coordinates of the turning point on the curve.
2. Sketch the curve, stating the coordinates of the points where it cuts the axes, and showing clearly its asymptotes and the turning point.

9 The cubic polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are real, is denoted by \(\mathrm { p } ( x )\).

1. Give a reason why the equation \(\mathrm { p } ( x ) = 0\) has at least one real root.
2. Suppose that the curve with equation \(y = \mathrm { p } ( x )\) has a local minimum point and a local maximum point with \(y\)-coordinates \(y _ { \text {min } }\) and \(y _ { \text {max } }\) respectively.
   1. Prove that if \(y _ { \text {min } } y _ { \text {max } } < 0\), then the equation \(\mathrm { p } ( x ) = 0\) has three real roots.
   2. Comment on the number of distinct real roots of the equation \(\mathrm { p } ( x ) = 0\) in the case \(y _ { \text {min } } y _ { \text {max } } = 0\).
   3. Suppose instead that the equation \(\mathrm { p } ( x ) = 0\) has only one real root for all values of \(c\). Prove that \(a ^ { 2 } \leqslant 3 b\).
   4. The iterative scheme $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 1 } { 3 x _ { n } ^ { 2 } + 1 } , \quad x _ { 0 } = 0$$ converges to a root of the cubic equation \(\mathrm { p } ( x ) = 0\).
      (a) Find \(\mathrm { p } ( x )\).
      (b) Find the limit of the iteration, correct to 4 decimal places.
   5. Determine the rate of convergence of the iterative scheme.

9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant.

1. Obtain the equation of each of the asymptotes of \(C\).
2. Find the coordinates of the turning points of \(C\).
3. In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\).

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 \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x\).

A rectangle is inscribed in a semicircle with centre \(O\) and radius 4. The point \(P ( x , y )\) is the vertex of the rectangle in the first quadrant as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area \(A\) of the rectangle as a function of \(x\).
b) Show that the maximum value of \(A\) occurs when \(y = x\).

The equation of a curve is $$y = k\sqrt{4x + 1} - x + 5,$$ where \(k\) is a positive constant.

1. Find \(\frac{dy}{dx}\). [2]
2. Find the \(x\)-coordinate of the stationary point in terms of \(k\). [2]
3. Given that \(k = 10.5\), find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of \(\tan^{-1}(2)\) with the positive \(x\)-axis. [4]

The curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\). \begin{enumerate}[label=(\alph*)] \item Find the coordinates of \(A\) and \(B\). [4] \end enumerate}

\includegraphics{figure_6} The diagram shows the curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) and the line \(AB\). It is given that the equation of \(AB\) is \(y = \frac{2x-32}{3}\). Find the area of the shaded region between the curve and the line. [5]

The equation of a curve is \(y = f(x)\), where \(f(x) = (4x - 3)^{\frac{3}{4}} - \frac{20}{3}x\).

1. Find the \(x\)-coordinates of the stationary points of the curve and determine their nature. [6]
2. State the set of values for which the function f is increasing. [1]

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 \(y = 4 + 5x + 6x^2 - 3x^3\).

1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases. [4]
2. It is given that \(y = 9x + k\) is a tangent to the curve. Find the value of the constant \(k\). [4]

A function f with domain \(x > 0\) is such that \(\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}\). It is given that the curve with equation \(y = \mathrm{f}(x)\) passes through the point \((1, 0)\).

1. Find the equation of the normal to the curve at the point \((1, 0)\). [3]
2. Find f\((x)\). [4]

It is given that the equation \(\mathrm{f}'(x) = 0\) can be expressed in the form $$125x^2 - 128x + 192 = 0.$$

1. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]

The equation of a curve is \(y = kx^{\frac{1}{2}} - 4x^2 + 2\), where \(k\) is a constant.

1. Find \(\frac{\text{d}y}{\text{d}x}\) and \(\frac{\text{d}^2y}{\text{d}x^2}\) in terms of \(k\). [2]
2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature. [4]
3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6. Find this value of \(k\). [6]

A solid rectangular block has a square base of side \(x\) cm. The height of the block is \(h\) cm and the total surface area of the block is \(96\) cm\(^2\).

1. Express \(h\) in terms of \(x\) and show that the volume, \(V\) cm\(^3\), of the block is given by $$V = 24x - \frac{1}{2}x^3.$$ [3]

Given that \(x\) can vary,

1. find the stationary value of \(V\), [3]
2. determine whether this stationary value is a maximum or a minimum. [2]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 4\sqrt{x} - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).

1. Find the coordinates of \(A\) and \(M\). [5]
2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis, giving your answer in terms of \(\pi\). [6]

\includegraphics{figure_9} The diagram shows part of the curve \(y = -x^2 + 8x - 10\) which passes through the points \(A\) and \(B\). The curve has a maximum point at \(A\) and the gradient of the line \(BA\) is \(2\).

1. Find the coordinates of \(A\) and \(B\). [7]
2. Find \(\int y \, dx\) and hence evaluate the area of the shaded region. [4]

The function \(f\) is such that \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\).

1. Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\). [3]

The function \(g\) is such that \(g(x) = 8 - (x - 2)^2\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which \(g\) has an inverse. [1]

For this value of \(k\),

1. find an expression for \(g^{-1}(x)\), [3]
2. sketch, on the same diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\). [3]

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]

Variables \(u\), \(x\) and \(y\) are such that \(u = 2x(y - x)\) and \(x + 3y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\). [5]

The line \(3y + x = 25\) is a normal to the curve \(y = x^2 - 5x + k\). Find the value of the constant \(k\). [6]

A curve is such that \(\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b\). The curve has stationary points at \((-1, 2)\) and \((3, k)\). Find the values of the constants \(a\), \(b\) and \(k\). [8]

A curve has equation \(y = (2x - 1)^{-1} + 2x\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]

\includegraphics{figure_10} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}}\).

1. The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(AB\). [6]
2. The tangent to the curve at a point \(T\) is parallel to \(AB\). Find the coordinates of \(T\). [3]
3. Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(AB\). [3]

A curve \(y = \mathrm{f}(x)\) has a stationary point at \(P(3, -10)\). It is given that \(\mathrm{f}'(x) = 2x^2 + kx - 12\), where \(k\) is a constant.

1. Show that \(k = -2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\). [4]
2. Find \(\mathrm{f}''(x)\) and determine the nature of each of the stationary points \(P\) and \(Q\). [2]
3. Find \(\mathrm{f}(x)\). [4]

The equation of a curve is \(y = x^3 + ax^2 + bx\), where \(a\) and \(b\) are constants.

1. In the case where the curve has no stationary point, show that \(a^2 < 3b\). [3]
2. In the case where \(a = -6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\). [3]