# 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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