1.07n Stationary points: find maxima, minima using derivatives

\includegraphics{figure_12} The figure above shows the curve \(C\) with equation $$f(x) = \frac{x+4}{\sqrt{x}}, \quad x > 0.$$

1. Determine the coordinates of the minimum point of \(C\), labelled as \(M\). [5]

The point \(N\) lies on the \(x\) axis so that \(MN\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(MN\) and the straight line with equation \(x = 1\).

1. Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\). [3]

Given the function \(f(x) = x - x^2\), defined for all real values of \(x\),

1. Show that \(f'(x) = 1 - 2x\) by differentiating \(f(x)\) from first principles. [4]
2. Find the maximum value of \(f(x)\). [1]
3. Explain why \(f^{-1}(x)\) does not exist. [1]

The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). \includegraphics{figure_1} The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is 48 units\(^2\), determine the \(y\)-coordinate of \(P\). [7]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]

The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$

1. Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a^b\) where \(a\) and \(b\) are rational numbers. [4]
2. Find the nature of the stationary point, justifying your answer. [2]

\includegraphics{figure_1} Figure 1 shows a sketch of a curve C with equation \(y = \text{f}(x)\), where f(x) is a quartic expression in \(x\). The curve • has maximum turning points at \((-1, 0)\) and \((5, 0)\) • crosses the \(y\)-axis at \((0, -75)\) • has a minimum turning point at \(x = 2\)

1. Find the set of values of \(x\) for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
2. Find the equation of C. You may leave your answer in factorised form. [3]

The curve \(C_1\) has equation \(y = \text{f}(x) + k\), where \(k\) is a constant. Given that the graph of \(C_1\) intersects the \(x\)-axis at exactly four places,

1. find the range of possible values for \(k\). [2]

The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_1}

1. The greatest value of \(r\) on the curve occurs at the point \(P\).
   1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
   2. Find the value of \(r\) at the point \(P\). [1]
   3. Mark the point \(P\) on a copy of the graph. [1]
2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]

In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin\theta}e^{\cos\theta}\) for \(0 \leq \theta < \pi\). \includegraphics{figure_5}

1. Find the exact area enclosed by the curve. [4]
2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}}e^{\frac{1}{2}}\). [7]

In this question you must show detailed reasoning.

1. Find the coordinates of all stationary points on the graph of \(y = 6\sinh^2 x - 13\cosh x\), giving your answers in an exact, simplified form. [9]
2. By finding the second derivative, classify the stationary points found in part (i). [3]

In this question you must show detailed reasoning. \includegraphics{figure_5} The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm{f}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ The diagram shows the curve \(y = \mathrm{f}(x)\).

1. Find the range of f. [6]

1. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm{g}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ Given that g is a one-one function, state the least possible value of \(k\). [1]

1. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis. [7]

A curve has equation \(y = x^5 - 5x^4\).

1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
2. Verify that the curve has a stationary point when \(x = 4\). [2]
3. Determine the nature of this stationary point. [2]

A curve has equation \(y = x^2 + kx - 4x^{-1}\) where \(k\) is a constant. Given that the curve has a minimum point when \(x = -2\)

• find the value of \(k\)
• show that the curve has a point of inflection which is not a stationary point. [7]

Let \(y = (x - 1)\left(\frac{2}{x^2} + t\right)\) define \(y\) as a function of \(x\) (\(x > 0\)), for each value of the real parameter \(t\).

1. When \(t = 0\),
   1. determine the set of values of \(x\) for which \(y\) is positive and an increasing function, [3]
   2. locate the stationary point of \(y\), and determine its nature. [2]
2. It is given that \(t = 2\) and \(y = -2\).
   1. Show that \(x\) satisfies \(f(x) = 0\), where \(f(x) = x^3 + x - 1\). [1]
   2. Prove that \(f\) has no stationary points. [2]
   3. Use the Newton-Raphson method, with \(x_0 = 1\), to find \(x\) correct to 4 significant figures. [4]

The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).

1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
3. 1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
   2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]

\includegraphics{figure_9} The diagram shows a sector of a circle, \(OMN\). The angle \(MON\) is \(2x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and perimeter, \(P\), of the sector. [2]
2. Given that \(P = 20\), show that \(A = \frac{100x}{(1 + x)^2}\). [2]
3. Find \(\frac{dA}{dx}\), and hence find the value of \(x\) for which the area of the sector is a maximum. [5]

The function f is defined by \(f : t \mapsto 2 \sin t + \cos 2t\) for \(0 \leqslant t < 2\pi\).

1. Show that \(\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)\). [2]
2. Determine the range of f. [5]

A curve \(C\) is given parametrically by \(x = 2 \cos t + \sin 2t\), \(y = f(t)\) for \(0 \leqslant t < 2\pi\).

1. Show that \(x^2 + y^2 = 5 + 4 \sin 3t\). [3]
2. Deduce that \(C\) lies between two circles centred at the origin, and touches both. [2]
3. Find the gradient of the tangent to \(C\) at the point at which \(t = 0\). [3]

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

1. Write down the equation of the asymptote of \(C\) and the coordinates of any points where \(C\) meets the coordinate axes. [2]
2. Show that the curve meets the line \(y = k\) if and only if \(-1 \leqslant k \leqslant 1\). Deduce the coordinates of the turning points of the curve. [5]

[Note: You are NOT required to sketch \(C\).]

A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).

1. Determine the equations of the asymptotes. [3]
2. Find the coordinates of the turning points. [5]
3. Sketch the curve. [3]

The equation of a curve is \(y = x^{\frac{3}{2}} \ln x\). Find the exact coordinates of the stationary point on the curve. [5]

The curve \(C\) has parametric equations $$x = \cos^2 t$$ $$y = \cos t \sin t$$ where \(0 \leq t < \pi\)

1. Show that \(C\) is a circle and find its centre and its radius. [5]

% Figure 1 shows a sketch of C with point P, rectangle R with diagonal OP \includegraphics{figure_1} Figure 1 Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \((\cos^2 \alpha, \cos\alpha \sin \alpha)\), \(0 < \alpha < \frac{\pi}{2}\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(OP\) as a diagonal, where \(O\) is the origin.

1. Show that the area of \(R\) is \(\sin\alpha \cos^3 \alpha\) [1]
2. Find the maximum area of \(R\), as \(\alpha\) varies. [7]

[Total 13 marks]

% Figure 4 shows curves with asymptotic behavior at x = 3 \includegraphics{figure_4} Figure 4

1. Figure 4 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 - 5}{3-x}, \quad x \in \mathbb{R}, x \neq 3$$ The curve has a minimum at the point \(A\), with \(x\)-coordinate \(\alpha\), and a maximum at the point \(B\), with \(x\)-coordinate \(\beta\). Find the value of \(\alpha\), the value of \(\beta\) and the \(y\)-coordinates of the points \(A\) and \(B\). [5]
2. The functions \(g\) and \(h\) are defined as follows $$g: x \to x + p \quad x \in \mathbb{R}$$ $$h: x \to |x| \quad x \in \mathbb{R}$$ where \(p\) is a constant. % Figure 5 shows curve with minimum points at C and D symmetric about y-axis \includegraphics{figure_5} Figure 5 Figure 5 shows a sketch of the curve with equation \(y = h(fg(x) + q)\), \(x \in \mathbb{R}\), \(x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).
   1. Find the value of \(p\) and the value of \(q\).
   2. Write down the coordinates of \(D\).
   [5]
3. The function \(\mathrm{m}\) is given by $$\mathrm{m}(x) = \frac{x^2 - 5}{3-x} \quad x \in \mathbb{R}, x < \alpha$$ where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
   1. Find \(\mathrm{m}^{-1}\)
   2. Write down the domain of \(\mathrm{m}^{-1}\)
   3. Find the value of \(t\) such that \(\mathrm{m}(t) = \mathrm{m}^{-1}(t)\)
   [10]

[Total 20 marks]

% Figure shows a curve with maximum at point A, passing through origin O, with horizontal asymptote \includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\) where $$f(x) = \frac{x^2 + 16}{3x} \quad x \neq 0$$ The curve has a maximum at the point \(A\) with coordinates \((a, b)\).

1. Find the value of \(a\) and the value of \(b\). [4] The function g is defined as $$g : x \mapsto \frac{x^2 + 16}{3x} \quad a \leq x < 0$$ where \(a\) is the value found in part (a).
2. Write down the range of g. [1]
3. On the same axes sketch \(y = g(x)\) and \(y = g^{-1}(x)\). [3]
4. Find an expression for \(g^{-1}(x)\) and state the domain of \(g^{-1}\) [5]
5. Solve the equation \(g(x) = g^{-1}(x)\). [3]