For which values of x is f(x) = x^3 - 6x^2 + 9x + 1 strictly increasing?

Worked solution for TMUA-P1-MM6-010 is revealed on this page behind the solution details.

Why isn't the answer to TMUA-P1-MM6-010 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P1-MM6-010 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P1-MM6-010?

No — calculators are banned in the TMUA. TMUA-P1-MM6-010 is set for a no-calculator target of about 240 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P1-MM6-010?

TMUA-P1-MM6-010 is Paper P1 · Differentiation (specification code MM6), level 3. Drill more of the same topic from this page.

TMUA-P1-MM6-010 — Paper P1 · Differentiation · Level 3

P1MM6Level 3~240s target

Spec: MM6 · No calculator · ~240s

For which values of

x

f(x) = x^3 - 6x^2 + 9x + 1

strictly increasing?

Options

$x < 1$ or $x > 3$
$1 < x < 3$
$x > 1$
$x < 3$
all real $x$

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: increasing-decreasing, between-roots-trap. Name the trap first, work the stem by hand, then open the solution below.

Show solution

f'(x) = 3x^2 - 12x + 9 = 3(x-1)(x-3) > 0

outside the roots, i.e.

x < 1

x > 3

. Trap:

1 < x < 3

is where

f

is decreasing — plausible since it is the interval between the stationary points.

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Keep practising

Same topic · same level

P1MM6Level 3

TMUA-P1-MM6-009

The area of a circle is A = π r^2. The radius increases at 2 cm/s. At what rate is the area increasing when r = 5 cm?

Differentiation

Open TMUA-P1-MM6-009 →

Same trap: increasing-decreasing

P1MM6Level 2

TMUA-P1-MM6-020

For f(x) = x^3 - 3x, on which interval is f strictly decreasing?

Differentiation

Open TMUA-P1-MM6-020 →

TMUA-P1-MM6-010 — Paper P1 · Differentiation · Level 3

Options

Keep practising

Master Differentiation

TMUA-P1-MM6-010 — Paper P1 · Differentiation · Level 3

Options

Keep practising

Master Differentiation