(a) Two \( x \)-intercepts for \( a = 3 \):

For \( y = x^3 + 3x^2 + b \), set \( y = 0 \): \( x^3 + 3x^2 + b = 0 \quad (M1) \).

Derivative: \( \frac{dy}{dx} = 3x^2 + 6x = 3x(x + 2) \), critical points at \( x = 0, -2 \quad (A1) \).

Exactly two intercepts occur when the cubic has a repeated root (discriminant condition or critical point analysis). Test \( b = -4 \): \( y = x^3 + 3x^2 – 4 \), roots at \( x = 1, -2 \) (double root). Test \( b = 0 \): \( y = x^3 + 3x^2 \), roots at \( x = 0, -3 \). Thus, \( b = -4, 0 \quad (A1) \).

[3 marks]

(b) Number of \( x \)-intercepts for \( a = 3 \):

(i) One intercept: \( b < -4 \) or \( b > 0 \quad (A1) \).

[1 mark]

(ii) Three intercepts: \( -4 < b < 0 \quad (A1) \).

[1 mark]

(c) Number of \( x \)-intercepts for \( a = -3 \):

For \( y = x^3 – 3x^2 + b \):

(i) Two intercepts: \( b = 0, 4 \quad (A1) \).

[1 mark]

(ii) One intercept: \( b < 0 \) or \( b > 4 \quad (A1) \).

[1 mark]

(iii) Three intercepts: \( 0 < b < 4 \quad (A1) \).

[1 mark]

(d) Points of Zero Gradient for Three Intercepts:

One point of zero gradient is above the \( x \)-axis, and one is below (A1).

[1 mark]

(e) Points of Zero Gradient:

\[ \frac{dy}{dx} = 3x^2 + 2ax = x(3x + 2a) = 0 \implies x = 0, -\frac{2}{3}a \quad (M1) \]

For \( x = 0 \): \( y = b \), so \( P(0, b) \quad (A1) \).

For \( x = -\frac{2}{3}a \): \( y = \left(-\frac{2}{3}a\right)^3 + a\left(-\frac{2}{3}a\right)^2 + b = -\frac{8}{27}a^3 + \frac{4}{9}a^3 + b = \frac{4}{27}a^3 + b \), so \( Q\left(-\frac{2}{3}a, \frac{4}{27}a^3 + b\right) \quad (A1) \).

[3 marks]

(f) Analysis for \( a > 0 \), \( b > 0 \):

(i) Second derivative: \( \frac{d^2y}{dx^2} = 6x + 2a \quad (M1) \).

At \( x = 0 \): \( \frac{d^2y}{dx^2} = 2a > 0 \), so \( P \) is a local minimum. At \( x = -\frac{2}{3}a \): \( \frac{d^2y}{dx^2} = -2a < 0 \), so \( Q \) is a local maximum (A1).

[2 marks]

(ii) \( P(0, b) \): \( y = b > 0 \). \( Q\left(-\frac{2}{3}a, \frac{4}{27}a^3 + b\right) \): \( \frac{4}{27}a^3 + b > 0 \) since \( a, b > 0 \). Both above the \( x \)-axis (A1).

[1 mark]

(g) Analysis for \( a < 0 \), \( b > 0 \):

(i) At \( x = 0 \): \( \frac{d^2y}{dx^2} = 2a < 0 \), so \( P \) is a local maximum; \( y = b > 0 \), so above the \( x \)-axis (A1).

[1 mark]

(ii) \( Q \): Below the \( x \)-axis when \( \frac{4}{27}a^3 + b < 0 \quad (A1) \).

[1 mark]

(h) Prove Three Intercepts:

Factorize: \( 4a^3b + 27b^2 = b(4a^3 + 27b) < 0 \quad (M1) \).

Requires \( b \) and \( \frac{4}{27}a^3 + b \) to have opposite signs, placing \( P \) and \( Q \) on opposite sides of the \( x \)-axis (A1).

This implies three \( x \)-intercepts, as the cubic crosses the \( x \)-axis between, before, and after the critical points (A1).

[3 marks]