2016 VCE Maths Methods Mini Test 4

Question 1 [2016 Exam 2 Section B Q2]

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

a.

i. Given that \(g'(x) = f(x)\) and \(g(0) = 1\), show that \(g(x) = -\frac{x^4}{12} + \frac{x^2}{2} - \frac{2x}{3} + 1\). 1 mark

ii. Find the values of \(x\) for which the graph of \(y = g(x)\) has a stationary point. 1 mark

b. The diagram below shows part of the graph of \(y = g(x)\), the tangent to the graph at \(x = 2\) and a straight line drawn perpendicular to the tangent to the graph at \(x = 2\). The equation of the tangent at the point \(A\) with coordinates \((2, g(2))\) is \(y = 3 - \frac{4x}{3}\).
The tangent cuts the \(y\)-axis at \(B\). The line perpendicular to the tangent cuts the \(y\)-axis at \(C\).

i. Find the coordinates of \(B\). 1 mark

ii. Find the equation of the line that passes through \(A\) and \(C\) and, hence, find the coordinates of \(C\). 2 marks

iii. Find the area of triangle \(ABC\). 2 marks

c. The tangent at \(D\) is parallel to the tangent at \(A\). It intersects the line passing through \(A\) and \(C\) at \(E\).

i. Find the coordinates of \(D\). 2 marks

ii. Find the length of \(AE\). 3 marks