1.07p Points of inflection: using second derivative

The diagram below shows a sketch of the graph of \(y = f'(x)\) for the interval \([x_1, x_5]\). \includegraphics{figure_13}

1. Find the interval on which \(f(x)\) is both decreasing and convex. Give reasons for your answer. [2]
2. Write down the \(x\)-coordinate of a point of inflection of the graph of \(y = f(x)\). [1]

The curve \(y = ax^4 + bx^3 + 18x^2\) has a point of inflection at \((1, 11)\).

1. Show that \(2a + b + 6 = 0\). [2]
2. Find the values of the constants \(a\) and \(b\) and show that the curve has another point of inflection at \((3, 27)\). [8]
3. Sketch the curve, identifying all the stationary points including their nature. [6]

A curve has equation \(y = g(x)\). Given that • \(g(x)\) is a cubic expression in which the coefficient of \(x^3\) is equal to the coefficient of \(x\) • the curve with equation \(y = g(x)\) passes through the origin • the curve with equation \(y = g(x)\) has a stationary point at \((2, 9)\)

1. find \(g(x)\), [7]
2. prove that the stationary point at \((2, 9)\) is a maximum. [2]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.

1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
2. find, showing your working, the integer value of \(k\). [5]

1. Show that the two non-stationary points of inflection on the curve \(y = \ln(1 + 4x^2)\) are at \(x = \pm\frac{1}{2}\). [6]

\includegraphics{figure_9} The diagram shows the curve \(y = \ln(1 + 4x^2)\). The shaded region is bounded by the curve and a line parallel to the \(x\)-axis which meets the curve where \(x = \frac{1}{2}\) and \(x = -\frac{1}{2}\).

1. Show that the area of the shaded region is given by $$\int_0^{\ln 2} \sqrt{e^y - 1} \, dy.$$ [3]
2. Show that the substitution \(e^y = \sec^2\theta\) transforms the integral in part (ii) to \(\int_0^{\frac{\pi}{4}} 2\tan^2\theta \, d\theta\). [2]
3. Hence find the exact area of the shaded region. [3]

A curve has equation \(y = xe^{\frac{x}{2}}\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]

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]

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]