## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | IB Analysis and Approaches |

Consider the cubic function \(f(x)=\frac{1}{6}x^3-2x^2+6x-2\)

(a) Find \(f'(x)\)

The graph of \(f\) has horizontal tangents at the points where \(x = a\) and \(x = b\) where \( a < b \).

(b) Find the value of \(a\) and the value of \(b\)

(c) Sketch the graph of \(y = f'(x)\).

(d) Hence explain why the graph of \(f\) has a local maximum point at \(x = a\).

(e) Find \(f''(b) \).

(f) Hence, use your answer to part (e) to show that the graph of \(f\) has a local minimum point at \(x = b\).

(g) Find the coordinates of the point where the normal to the graph of \(f\) at \(x = a\) and the tangent to the graph of \(f\) at \(x = b\) intersect.

## 2. | IB Standard |

The following diagram shows part of the graph of \(y=f (x)\)

The graph has a local maximum where \(x=- \frac23\), and a local minimum where \(x=4\).

sketch the graph of \(y=f'(x)\) for \(-4\le x \le 7\)

Write down the following in order from least to greatest: \(f(2),f'(4)\) and \(f''(4)\).

## 3. | IB Analysis and Approaches |

Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \).

(a) Find an expression for the first derivative \(f'(x) \).

The graph of \(f\) has exactly one maximum point at P.

(b) Find the x-coordinate of P.

The graph of \(f\) has exactly one point of inflection at Q.

(c) Find the x-coordinate of Q.

(d) The region enclosed by the graph of \(f\), the x-axis, and the vertical lines through P and Q has an area of one square unit, find the value of \(k\).

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