ARCH-MM-09: Stationary Point Classification, Monotonic Cubics & Constrained 2D/3D Optimisation

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-09: Why the 175s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~175s (Blows 90s Clock)

Differentiating gives

\frac{dy}{dx} = 3x^2 + 2px + q = 0

x = 1

and

x = 3

. Setting up simultaneous equations

3 + 2p + q = 0

and

27 + 6p + q = 0

and solving gives

p = -6

and

q = 9

, so

p + q = 3

(~140s).

The Fast ShortcutTarget: 50s (Saves 125s)

Line 1 (Factorised Derivative Identity):

y'(x) = 3x^2 + 2px + q \equiv 3(x - 1)(x - 3) = 3x^2 - 12x + 9

.
**Line 2 (Read Off

p, q

):**

2p = -12 \implies p = -6,\; q = 9 \implies p + q = 3

(25s).

Where Students Lose Time: In 29 past UAT-UK questions (7.07% of Mathematics 2), candidates lose 125s by forgetting that f'(a)=0 and f''(a)=0 with no sign change in f'(x) is a point of inflection. Instead, apply MM6 with Maths 1 skill M5.11.

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