VCE Methods Integral Calculus Application Task 16

Question 1 [2018 Exam 2 Section B Q1]

Consider the quartic \( f : \mathbb{R} \rightarrow \mathbb{R}, f(x) = 3x^4 + 4x^3 - 12x^2 \) and part of the graph of \(y = f(x)\) below.

a. Find the coordinates of the point \(M\), at which the minimum value of the function \(f\) occurs. 1 mark

b. State the values of \(b \in \mathbb{R}\) for which the graph of \(y = f(x) + b\) has no x-intercepts. 1 mark

Part of the tangent, \(l\), to \(y = f(x)\) at \(x = -\frac{1}{3}\) is shown below.

c. Find the equation of the tangent \(l\). 1 mark

d. The tangent \(l\) intersects \(y = f(x)\) at \(x = -\frac{1}{3}\) and at two other points.
State the x-values of the two other points of intersection. Express your answers in the form \(\frac{a \pm \sqrt{b}}{c}\), where \(a, b\) and \(c\) are integers. 2 marks

e. Find the total area of the regions bounded by the tangent \(l\) and \(y = f(x)\). Express your answer in the form \(\frac{a\sqrt{b}}{c}\), where \(a, b\) and \(c\) are positive integers. 2 marks

Let \( p : \mathbb{R} \rightarrow \mathbb{R}, p(x) = 3x^4 + 4x^3 + 6(a-2)x^2 - 12ax + a^2, a \in \mathbb{R} \).

f. State the value of \(a\) for which \(f(x) = p(x)\) for all \(x\). 1 mark

g. Find all solutions to \(p'(x) = 0\), in terms of \(a\) where appropriate. 1 mark

h.

i. Find the values of \(a\) for which \(p\) has only one stationary point. 1 mark

ii. Find the minimum value of \(p\) when \(a = 2\). 1 mark

iii. If \(p\) has only one stationary point, find the values of \(a\) for which \(p(x) = 0\) has no solutions. 2 marks