1.07n Stationary points: find maxima, minima using derivatives

A curve \(y = f(x)\) has a stationary point at \((3, 7)\) and is such that \(f''(x) = 36x^{-3}\).

1. State, with a reason, whether this stationary point is a maximum or a minimum. [1]
2. Find \(f'(x)\) and \(f(x)\). [7]

A curve has equation \(y = (kx - 3)^{-1} + (kx - 3)\), where \(k\) is a non-zero constant.

1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point. Justify your answers. [7]

1. \includegraphics{figure_3} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360°\) about the \(x\)-axis. [5]

\includegraphics{figure_3} The diagram shows part of the curve \(y = x(9 - x^2)\) and the line \(y = 5x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.

1. Express the length of \(PQ\) in terms of \(t\), simplifying your answer. [2]
2. Given that \(t\) can vary, find the maximum value of the length of \(PQ\). [3]

\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).

1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
2. Find the coordinates of \(M\). [3]
3. Find, showing all necessary working, the area of the shaded region. [4]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{4e^{2x} + 9}{e^x + 2}\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).

1. Find the exact value of the gradient of the curve at \(P\). [4]
2. Find the exact coordinates of \(M\). [4]

\includegraphics{figure_6} The diagram shows the curve with parametric equations $$x = 1 + \sqrt{t}, \quad y = (\ln t + 2)(\ln t - 3),$$ for \(0 < t < 25\). The curve crosses the \(x\)-axis at the points \(A\) and \(B\) and has a minimum point \(M\).

1. Show that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4\ln t - 2}{\sqrt{t}}\). [4]
2. Find the exact gradient of the curve at \(B\). [2]
3. Find the exact coordinates of \(M\). [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]

A curve has parametric equations $$x = \ln(t + 1), \quad y = t^2 \ln t.$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Find the exact value of \(t\) at the stationary point. [2]
3. Find the gradient of the curve at the point where it crosses the \(x\)-axis. [2]

A curve has parametric equations $$x = t + \ln(t + 1), \quad y = 3te^{2t}.$$

1. Find the equation of the tangent to the curve at the origin. [5]
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

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_10} The diagram shows the curve \(y = \sin 2x \cos^2 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\).

1. Using the substitution \(u = \sin x\), find the exact area of the region bounded by the curve and the \(x\)-axis. [5]
2. Find the exact \(x\)-coordinate of \(M\). [6]

A balloon in the shape of a sphere has volume \(V\) and radius \(r\). Air is pumped into the balloon at a constant rate of \(40\pi\) starting when time \(t = 0\) and \(r = 0\). At the same time, air begins to flow out of the balloon at a rate of \(0.8\pi r\). The balloon remains a sphere at all times.

1. Show that \(r\) and \(t\) satisfy the differential equation $$\frac{dr}{dt} = \frac{50 - r}{5r^2}.$$ [3]
2. Find the quotient and remainder when \(5r^2\) is divided by \(50 - r\). [3]
3. Solve the differential equation in part (a), obtaining an expression for \(t\) in terms of \(r\). [6]
4. Find the value of \(t\) when the radius of the balloon is 12. [1]

\includegraphics{figure_8} The diagram shows a sketch of the curve \(y = x^2\ln x\) and its minimum point \(M\). The curve cuts the \(x\)-axis at the point \((1, 0)\).

1. Find the exact value of the \(x\)-coordinate of \(M\). [4]
2. Use integration by parts to find the area of the shaded region enclosed by the curve, the \(x\)-axis and the line \(x = 4\). Give your answer correct to 2 decimal places. [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]

The curve \(y = \frac{\ln x}{x + 1}\) has one stationary point.

1. Show that the \(x\)-coordinate of this point satisfies the equation $$x = \frac{x + 1}{\ln x},$$ and that this \(x\)-coordinate lies between 3 and 4. [5]
2. Use the iterative formula $$x_{n+1} = \frac{x_n + 1}{\ln x_n}$$ to determine the \(x\)-coordinate correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

The function f is defined by \(\mathrm{f} : x \mapsto 7 - 2x^2 - 12x\) for \(x \in \mathbb{R}\).

1. Express \(7 - 2x^2 - 12x\) in the form \(a - 2(x + b)^2\), where \(a\) and \(b\) are constants. [2]
2. State the coordinates of the stationary point on the curve \(y = \mathrm{f}(x)\). [1]

The function g is defined by \(\mathrm{g} : x \mapsto 7 - 2x^2 - 12x\) for \(x \geqslant k\).

1. State the smallest value of \(k\) for which g has an inverse. [1]
2. For this value of \(k\), find \(\mathrm{g}^{-1}(x)\). [3]

\includegraphics{figure_11} The diagram shows part of the curve \(y = (x + 1)^2 + (x + 1)^{-1}\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2(x + 1)^3 = 1\) and find the exact value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) at \(A\). [5]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]

\includegraphics{figure_9} The diagram shows the curve \(y = (1 + x^2)\text{e}^{-\frac{3x}{4}}\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).

1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve. [4]
2. Show that the exact value of the area of \(R\) is \(18 - \frac{42}{\text{e}}\). [5]

A curve has equation \(y = \frac{3 \cos x}{2 + \sin x}\), for \(-\frac{1}{2}\pi \leqslant x \leqslant \frac{1}{2}\pi\).

1. Find the exact coordinates of the stationary point of the curve. [6]
2. The constant \(a\) is such that \(\int_0^a \frac{3 \cos x}{2 + \sin x} \, dx = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures. [4]

\includegraphics{figure_7} The diagram shows the curve \(y = 5\sin^2 x \cos^3 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places. [5]
2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]

The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).

1. Find the equations of the asymptotes of \(C\). [2]
2. Find the coordinates of any stationary points on \(C\). [4]
3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]

Answer only one of the following two alternatives. EITHER The curve \(C\) has equation \(y = \frac{5(x-1)(x+2)}{(x-2)(x+3)}\).

1. Express \(y\) in the form \(P + \frac{Q}{x-2} + \frac{R}{x+3}\). [3]
2. Show that \(\frac{dy}{dx} = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\). [4]
3. Write down the equations of all the asymptotes of \(C\). [3]
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\). [4]

OR A curve has equation \(y = \frac{5}{3}x^{\frac{3}{2}}\), for \(x \geq 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).

1. Find the length of \(R\). [4]
2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\). [5]
3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac{232\pi}{15}\). [5]

The curve \(C\) has equation \(y = \frac{2x^2 + kx}{x + 1}\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. [5] For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes. [6]