← Unit 5

Unit 5 Cheat Sheet

Every test in Analytical Applications of Differentiation — and, more importantly, which test, and when. The whole unit runs on two signs: \(f'\) is the direction, \(f''\) is the bend. The look-alikes that leak points (critical point vs. extremum, \(f''=0\) vs. inflection, "where the max occurs" vs. "the max value") each get the decision that picks the right move — and the whole-unit skill, justifying like an AP reader, gets its own panel at the bottom.
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The Existence Theorems 5.1 · 5.2

Before you hunt for a max, a min, or a special tangent, two theorems promise the quarry is actually there.

Mean Value Theorem (instantaneous = average)
hypotheses — both are load-bearing
Rolle (MVT with a level secant)
critical point of f

EVT: a continuous function on a closed interval attains an absolute max and min — each at a critical point or an endpoint.

Which guarantee?
"some slope = avg"MVT — needs cont. on \([a,b]\), diff. on \((a,b)\)
"\(f'(c)=0\) exists"Rolle, when \(f(a)=f(b)\)
"abs max/min exist"EVT — continuous on a closed interval
find abs extremacandidates test: evaluate \(f\) at every critical point and both endpoints; pick largest/smallest
trap
  • MVT needs continuity AND differentiability — a corner (like \(|x|\)) voids it even though it's continuous.
  • The guaranteed \(c\) is interior — in \((a,b)\), never at an endpoint.
  • Critical point \(=f'=0\) OR \(f'\) DNE — cusps and corners count.

Increasing, Decreasing & Extrema 5.3 · 5.4

The sign of \(f'\) is the direction of travel. Where that sign flips sits a peak or a valley.

direction from the first derivative
first-derivative test (sign change of f′)
Classify a critical point c
\(+ \to -\)\(f\) rises then falls ⇒ local maximum
\(- \to +\)\(f\) falls then rises ⇒ local minimum
no sign changenot an extremum — just a shelf
trap
  • \(f'=0\) is only a candidate. No sign change ⇒ no extremum (e.g. \(x^3\)).
  • Don't skip \(f'\)-DNE points — cusps/corners can be extrema (\(x^{2/3}\)).
  • Increasing/decreasing is \(f'\); concavity is \(f''\) — don't cross the wires.

Concavity & the 2nd-Derivative Test 5.6 · 5.7

The sign of \(f''\) is the bend — cup up or dome down — and where it flips is an inflection point.

concavity from the second derivative
second-derivative test (at f′(c)=0)
inflection point
2nd-derivative test at a critical point
\(f''(c)<0\)concave down (dome) ⇒ local max
\(f''(c)>0\)concave up (cup) ⇒ local min
\(f''(c)=0\)inconclusive — fall back to the first-derivative test
trap
  • \(f''=0\) isn't automatically an inflection — it must change sign (\(x^4\) doesn't).
  • If \(f''(c)=0\), the test is silent — don't declare a max/min from it.
  • Concave up = holds water \(\smile\) = a min at a flat spot.

Connecting f, f′ & f″ 5.8 · 5.9

One curve, three views. A feature in one lane is a zero — or an extremum — in the next.

the chain of zeros
inflection ⇔ extremum of f′
Reading the graph of f′
above / below axis\(f'>0\Rightarrow f\nearrow\); \(f'<0\Rightarrow f\searrow\)
crosses the axis\(f\) has an extremum (read the crossing direction)
\(f'\) turns (max/min)\(f\) has an inflection point
\(f'\) rising / falling\(f\) concave up / down
trap
  • A peak of \(f'\) is an inflection of \(f\), not a max of \(f\).
  • "\(f'\) positive but decreasing" ⇒ \(f\) still rising (just concave down).
  • Concavity is where \(f'\) is increasing/decreasing — not where \(f'\) is \(+/-\).

Optimization 5.5 · 5.10 · 5.11

Turn a real question into one function; the derivative finds the single exact best. Everything above pays off here.

  1. Picture & name the variables from the words.
  2. Objective: write the quantity to maximize or minimize.
  3. Constraint kills a variable: solve the fixed fact for one variable and substitute — two variables become one.
  4. Domain: physical limits (lengths \(>0\); a cut \(<\) half the sheet).
  5. \(f'=0\): find the critical point(s).
  6. Justify & answer the question asked, with units.
Seal the winner
only crit. pt, \(f''<0\)it's the absolute maximum (\(>0\) ⇒ absolute min)
closed intervalcandidates test — compare critical points and endpoints
fixed perimeter
max area ⇒ square
fixed sum
max product ⇒ equal
fixed product
min sum ⇒ equal
cheapest can
min surface ⇒ \(h=2r\)
trap
  • Two variables ⇒ can't differentiate yet. Use the constraint first.
  • Check the domain: reject impossible critical points; on a closed interval, test the endpoints too.
  • Answer the question, with units — the area/volume/cost, not just \(x=25\).

Read the two signs, name the shape the curve decoder

Direction is \(f'\); bend is \(f''\). Two signs, four shapes — and this is the exact language "describe the graph" and "justify" questions want back.

\(f'>0,\ f''>0\)
Increasing, concave up
rising and bending upward — climbing faster (\(\nearrow\ \smile\))
\(f'>0,\ f''<0\)
Increasing, concave down
rising but leveling off — approaching a peak (\(\nearrow\ \frown\))
\(f'<0,\ f''<0\)
Decreasing, concave down
falling and dropping ever faster (\(\searrow\ \frown\))
\(f'<0,\ f''>0\)
Decreasing, concave up
falling but slowing its fall — approaching a valley (\(\searrow\ \smile\))

The zeros tell the turns: \(f'=0\) (with a sign change) is a peak or valley of \(f\); \(f''=0\) (with a sign change) is an inflection — and that's also where \(f'\) itself peaks or bottoms out.

Justify like an AP reader the whole-unit skill

Unit 5 answers are graded on the because. Every claim below scores only when it names the derivative fact that forces it — the number alone earns nothing.

increasing / decreasing
"\(f\) is increasing on \((a,b)\) because \(f'>0\) there."
local max / min
"a local max at \(x=c\) because \(f'\) changes \(+\to-\) (or \(f''(c)<0\))."
concavity
"concave up on \((a,b)\) because \(f''>0\) there."
inflection point
"an inflection at \(x=c\) because \(f''\) changes sign there."
absolute extremum
"the abs max is \(f(c)\) because it's the largest among the critical points and endpoints."
optimization winner
"the max because \(x=c\) is the only critical point and \(f''(c)<0\)."
Before you write the answer
name the factevery "increasing / max / concave / inflection" claim cites the sign of \(f'\) or \(f''\) — explicitly
show the changeextrema and inflections need a sign change, not just a zero
answer the askgive the quantity requested (the value, the area) with units — reread the prompt

The reliable move: after any calculation, ask "which derivative sign forces this, and did it change sign?" State that clause and the justification points are automatic.

∴

Unit 5 is where the derivative becomes sight. Two theorems promise the peaks and valleys exist; \(f'\) tells you where they are and which way \(f\) travels; \(f''\) tells you how \(f\) bends and where it flips; connecting the three lets you read or draw any curve on sight; and optimization turns all of it into the single best answer. Master "\(f'\) is direction, \(f''\) is bend — and justify every claim" and the whole unit is one skill wearing five hats.

Test yourself on the exam set →
Existence Theorems Increasing & Decreasing Concavity Connecting f, f′, f″ Optimization