1.07f Convexity/concavity: points of inflection

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]

\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]