1.07o Increasing/decreasing: functions using sign of dy/dx

13 A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company models the logo on an \(x - y\) plane as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-20_492_492_511_776} Use calculus to find the maximum area of the rectangle.
Fully justify your answer.

10 Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } }\) is an increasing function for all values of \(x\).

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

8 The function f is given by \(\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 3 x ^ { 2 } - 1 }\), for \(x > 1\). Show that f is a decreasing function.

6

1. Find the coordinates of the stationary points of the curve with equation $$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$ and determine their nature.
2. Sketch the graph of \(y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }\) and hence state the set of values of \(k\) for which the equation \(3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k\) has exactly four distinct real roots.

5

1. Differentiate \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) with respect to \(x\).
2. Hence show that \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) is increasing for all \(x\).

\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.

1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]

A function f is such that \(f'(x) = 6(2x-3)^2 - 6x\) for \(x \in \mathbb{R}\).

1. Determine the set of values of \(x\) for which f\((x)\) is decreasing. [4]
2. Given that f\((1) = -1\), find f\((x)\). [4]

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 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 \(y = \frac{1}{6}(2x - 3)^3 - 4x\).

1. Find \(\frac{dy}{dx}\). [3]
2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis. [3]
3. Find the set of values of \(x\) for which \(\frac{1}{6}(2x - 3)^3 - 4x\) is an increasing function of \(x\). [3]

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]

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]

The function \(f\) is such that \(f(x) = x^3 - 3x^2 - 9x + 2\) for \(x > n\), where \(n\) is an integer. It is given that \(f\) is an increasing function. Find the least possible value of \(n\). [4]

\includegraphics{figure_3} Figure 3 shows the plan of a stage in the shape of a rectangle joined to a semicircle. The length of the stage is \(2x\) metres and the width is \(y\) metres. The diameter of the semicircular part is \(2x\) metres. The perimeter of the stage is 80 m.

1. Show that the area, \(A\) m², of the stage is given by $$A = 80x - \left(2 + \frac{\pi}{2}\right)x^2.$$ [4]
2. Use calculus to find the value of \(x\) at which \(A\) has a stationary value. [4]
3. Prove that the value of \(x\) you found in part (b) gives the maximum value of \(A\). [2]
4. Calculate, to the nearest m², the maximum area of the stage. [2]

\includegraphics{figure_1} Figure 1 shows part of a curve \(C\) with equation \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).

1. Find the exact area of \(R\). [8]
2. Use calculus to show that \(y\) is increasing for \(x > 2\). [4]

\includegraphics{figure_4} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle \(x\) metres by \(y\) metres. The height of the tank is \(x\) metres. The capacity of the tank is 100 m³.

1. Show that the area \(A\) m² of the sheet metal used to make the tank is given by $$A = \frac{300}{x} + 2x^2.$$ [4]
2. Use calculus to find the value of \(x\) for which \(A\) is stationary. [4]
3. Prove that this value of \(x\) gives a minimum value of \(A\). [2]
4. Calculate the minimum area of sheet metal needed to make the tank. [2]

A pencil holder is in the shape of an open circular cylinder of radius \(r\) cm and height \(h\) cm. The surface area of the cylinder (including the base) is 250 cm\(^2\).

1. Show that the volume, V cm\(^3\), of the cylinder is given by \(V = 125r - \frac{\pi r^3}{2}\). [4]
2. Use calculus to find the value of \(r\) for which \(V\) has a stationary value. [3]
3. Prove that the value of \(r\) you found in part (b) gives a maximum value for \(V\). [2]
4. Calculate, to the nearest cm\(^3\), the maximum volume of the pencil holder. [2]

\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 4. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by $$A = 4x^2 + \frac{3090}{x}.$$ [3]

The manufacturer needs to minimise the surface area of a carton.

1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
2. Calculate the minimum value of \(A\). [2]
3. Prove that this value of \(A\) is a minimum. [2]

\includegraphics{figure_9} A rectangular sheet of metal measures 50 cm by 40 cm. Squares of side \(x\) cm are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.

1. Show that the volume, \(V\) cm\(^3\), of the tray is given by $$V = 4x(x^2 - 45x + 500).$$ [3]
2. State the range of possible values of \(x\). [1]
3. Find the value of \(x\) for which \(V\) is a maximum. [4]
4. Hence find the maximum value of \(V\). [2]
5. Justify that the value of \(V\) you found in part (d) is a maximum. [2]

The function f, defined for \(x \in \mathbb{R}, x > 0\), is such that $$f'(x) = x^2 - 2 + \frac{1}{x^2}.$$

1. Find the value of \(f''(x)\) at \(x = 4\). [3]
2. Given that \(f(3) = 0\), find \(f(x)\). [4]
3. Prove that \(f\) is an increasing function. [3]