IB Mathematics AHL 2.12 Polynomial functions and their graphs AA HL Paper 3
Question 1
(a) Topic-AHL 2.12 Polynomial functions, their graphs and equations; zeros, roots and factors
(b) Topic-AHL 2.12 Polynomial functions, their graphs and equations; zeros, roots and factors
(c) Topic-AHL 2.12 Polynomial functions, their graphs and equations; zeros, roots and factors
(d) Topic-SL 5.1 Derivative interpreted as gradient function and as rate of change
(e) Topic-SL 5.1 Derivative interpreted as gradient function and as rate of change
(f) Topic-SL 5.8 Local maximum and minimum points.
(g) Topic-SL 5.8 Local maximum and minimum points.
(h) Topic-AHL 2.12 Polynomial functions, their graphs and equations; zeros, roots and factors
This question asks you to explore some properties of the family of curves \(y = x^3 + ax^2 + b \quad \text{where } x \in \mathbb{R} \text{ and } a, b \text{ are real parameters.}\)
Consider the family of curves \( y = x^3 + ax^2 + b \) for \( x \in \mathbb{R} \), where \( a \in \mathbb{R}, a \neq 0 \) and \( b \in \mathbb{R} \).
First consider the case where \( a = 3 \) and \( b \in \mathbb{R} \).
(a) By systematically varying the value of \( b \), or otherwise, find the two values of \( b \) such that the curve \( y = x^3 + 3x^2 + b \) has exactly two \( x \)-axis intercepts.
(b) Write down the set of values of \( b \) such that the curve \( y = x^3 + 3x^2 + b \) has exactly:
(i) one \( x \)-axis intercept;
(ii) three \( x \)-axis intercepts.
Now consider the case where \( a = -3 \) and \( b \in \mathbb{R} \).
(c) Write down the set of values of \( b \) such that the curve \( y = x^3 – 3x^2 + b \) has exactly:
(i) two \( x \)-axis intercepts;
(ii) one \( x \)-axis intercept;
(iii) three \( x \)-axis intercepts.
(d) Consider the case where the curve has exactly three \( x \)-axis intercepts. State whether each point of zero gradient is located above or below the \( x \)-axis.
(e) Show that the curve has a point of zero gradient at \( P(0, b) \) and a point of zero gradient at \(Q\left(-\frac{2}{3}a, \frac{4}{27}a^3 + b \right).\)
(f) Consider the points \( P \) and \( Q \) for \( a > 0 \) and \( b > 0 \).
(i) Find an expression for \( \frac{d^2y}{dx^2} \) and hence determine whether each point is a local maximum or a local minimum.
(ii) Determine whether each point is located above or below the \( x \)-axis.
(g) Consider the points \( P \) and \( Q \) for \( a < 0 \) and \( b > 0 \).
(i) State whether \( P \) is a local maximum or a local minimum and whether it is above or below the \( x \)-axis.
(ii) State the conditions on \( a \) and \( b \) that determine when \( Q \) is below the \( x \)-axis.
(h) Prove that if \( 4a^3b + 27b^2 < 0 \), then the curve \( y = x^3 + ax^2 + b \) has exactly three \( x \)-axis intercepts.
▶️Answer/Explanation
\(\textbf{1(a)}\)
varies the value of $b$ $\text{ with }$ $a=3$
\(b = −4, 0\)
\(\textbf{1(b)}\)
(i) \(b < −4\) or \(b > 0\)
(ii) \(−4<b < 0\)
\(\textbf{1(c)}\)
(i) \(b = 0, 4\)
(ii) \(b < 0 \quad \text{or} \quad b > 4\)
(iii) \(0 < b < 4\)
\(\textbf{1(d)}\)
one point of zero gradient is located on either side (of the \(x\)- axis) (or equivalent)
\(\textbf{1(e)}\)
\(\frac{dy}{dx} = 3x^2 + 2ax\)
Attempts to solve \(\frac{dy}{dx} = 0\) for \(x\):
\(
x(3x + 2a) = 0 \quad \text{OR} \quad x = \frac{-2a \pm \sqrt{4a^2}}{6} \quad \text{OR} \quad x + \frac{a}{3} = \pm \frac{a}{3}.
\)
\(
x = -\frac{2}{3}a, \, 0
\)
When \(x = 0\), \(y = b\), and so \(P(0, b)\) is a point of zero gradient.
Substitutes their expression for \( x \) in terms of \( a \) into \( y = x^3 + ax^2 + b \):
\(
y = \left(-\frac{2}{3}a\right)^3 + a\left(-\frac{2}{3}a\right)^2 + b
\)
\(
y = -\frac{8}{27}a^3 + \frac{4}{9}a^3 + b \quad \implies \quad y = -\frac{8}{27}a^3 + \frac{12}{27}a^3 + b
\)
\(
\text{So } Q\left(-\frac{2}{3}a, \frac{4}{27}a^3 + b\right) \text{ is a point of zero gradient.}
\)
\(\textbf{1(f)}\)
(i) \(\frac{d^2y}{dx^2} = 6x + 2a\)
When \( x = 0 \),
\(
\frac{d^2y}{dx^2} = 2a \quad (a > 0) \quad \text{and so } (P) \text{ is a (local) minimum (point)}.
\)
When \( x = -\frac{2}{3}a \),
\(
\frac{d^2y}{dx^2} = -2a \quad (a > 0) \quad \text{and so } (Q) \text{ is a (local) maximum (point)}.
\)
(ii) (\(P\) and \(Q\) are) both above (the \(x\)- axis)
\(\textbf{1(g)}\)
(i) ( \(P\) ) is a (local) maximum (point) and is above (the \(x\)- axis)
(ii) \((Q \text{ is below the } x\text{-axis when }) \quad \frac{4}{27}a^3 + b < 0\)
\(\textbf{1(h)}\)
Attempts to factorize \( 4a^3b + 27b^2 (< 0) \):
\(
27b\left(\frac{4}{27}a^3 + b\right) (< 0) \quad \text{OR} \quad b\left(4a^3 + 27b\right)(< 0).
\)
\(
b > 0 \text{ and } \frac{4}{27}a^3 + b < 0 \quad \text{or} \quad b < 0 \text{ and } \frac{4}{27}a^3 + b > 0.
\)
\(
\text{When } b \text{ and } \frac{4}{27}a^3 + b \text{ have opposite signs, } P \text{ and } Q \text{ are located on either side }
\)
\(
\text{(of the } x\text{-axis) (or equivalent)}.
\)
\(
P \text{ and } Q \text{ are located on either side (of the } x\text{-axis) if (and only if) the curve has exactly three } x\text{-axis intercepts.}
\)
\(
\text{If } 4a^3b + 27b^2 < 0, \text{ then the graph of } y = x^3 + ax^2 + b \text{ has exactly three } x\text{-axis intercepts.}
\)