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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 |
The following diagram shows the graph of \(f'\), the first derivative of a function \(f\).
The graph of \(f'\) has x-intercepts at \(x=a, x=c, x=e \text{ and } x=g\). It has local maximum points at \(x=b \text{ and } x=f \) and a local minimum point at \( x=d \).
(a) Find all the values of \(x\) where the graph of \(f\) is increasing. Justify your answer.
(b) Find all the values of \(x\) where the graph of \(f\) has a local maximum. Justify your answer.
(c) Find all the values of \(x\) where the graph of \(f\) has a local minimum. Justify your answer.
(d) Find all the values of \(x\) where the graph of \(f\) has points of inflection and determine which of these is a horizontal point of inflection.
(e) The total area of the region enclosed by graph of \(f'\) and the x-axis for \(a \lt x \lt e\) is 6.
Given that \( f(a) + f(e) = 3 \), find the value of \(f(c)\).
4. | 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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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
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