MAT148H1 Chap.9 Extreme Values, Indeterminate Forms and Curve Sketching
Extreme Values, Indeterminate Forms and Curve Sketching
Where maxima and minima can hide
An extreme value of f is a largest or smallest output, either across a whole domain (absolute or global) or only among nearby inputs (local). On a closed interval [a, b], a continuous function is guaranteed to reach both an absolute maximum and an absolute minimum by the Extreme Value Theorem. Derivatives tell you where to look.
If f is differentiable at an interior point c that is a local maximum or minimum, then f′(c) = 0. Points where f′(c) = 0, together with interior points where f′ does not exist, are called critical points. So every extremum on [a, b] sits at a critical point or at an endpoint, and the search reduces to evaluating f at a short list of candidates and comparing the numbers.
Being critical is necessary but not sufficient: f(x) = x³ has f′(0) = 0 while increasing straight through the origin. Notice that the comparison step needs no classification at all: a candidate that turns out not to be an extremum simply loses when the values are lined up.
Optimisation word problems follow the same pattern once a quantity such as area or volume is written as a function of one variable on a bounded interval of physically sensible inputs.
Classifying a critical point
The Mean Value Theorem links the sign of f′ to monotonicity: f′ > 0 on an interval forces f to increase there, and f′ < 0 forces it to decrease.
The First Derivative Test turns this into a classification rule. If f′ is positive just to the left of c and negative just to the right, c is a local maximum; the reverse pattern gives a local minimum. A sign chart, built by factoring f′ into linear pieces and tracking each sign, organises the work.
The Second Derivative Test is quicker when f″ is easy to compute: at a critical point, f″(c) < 0 signals a local maximum and f″(c) > 0 a local minimum. When f″(c) = 0 the test is silent, and you move to the first nonzero higher derivative: an odd order means no extremum, an even order means a maximum or minimum according to its sign.
Neither test applies at an endpoint, which is why absolute extremum questions rely on comparison instead.
Indeterminate forms and L'Hôpital's Rule
A quotient f(x)/g(x) is indeterminate of type 0/0 when both parts tend to zero, and of type ∞/∞ when both grow without bound in absolute value.
For exactly these types, L'Hôpital's Rule lets you replace the quotient by f′(x)/g′(x), provided f and g are differentiable near the point and the new limit exists (or g′ stays nonzero nearby). Each derivative is taken separately; nothing resembling the quotient rule is involved.
The rule may be applied repeatedly, but the indeterminate type must be rechecked before every application, since one round often produces a determinate limit that simple substitution finishes. Other shapes are converted first. A product of type 0 · ∞ becomes a quotient by moving one factor into the denominator as its reciprocal.
Exponential shapes 1^∞, 0⁰ and ∞⁰ are handled by taking a logarithm, computing the limit of the log, and exponentiating at the end.
Differences ∞ − ∞ usually yield after combining fractions over a common denominator.
Putting the information together in a sketch
Concavity describes whether f′ is increasing (concave up) or decreasing (concave down) near a point, so for twice differentiable functions it is read from the sign of f″.
An inflection point is a place where the concavity switches, which requires f″ = 0 or f″ undefined there, though neither condition alone guarantees a switch. A careful sketch collects domain, intercepts, symmetry, asymptotes, intervals of increase and decrease, extrema, and concavity, then draws a curve consistent with all of them.
Limits, sometimes evaluated with L'Hôpital's Rule, supply the asymptotic behaviour at the edges of the domain, while the two derivative tests fix the turning points and bends in the middle.
What this chapter covers
- 01
Critical points and the closed-interval method for absolute extrema
- 02
First Derivative Test with sign charts of f′
- 03
Second Derivative Test and higher-order checks when f″(c) = 0
- 04
L'Hôpital's Rule for 0/0 and ∞/∞, including repeated use
- 05
Rewriting 0 · ∞, ∞ − ∞ and exponential forms before applying the rule
- 06
Concavity, inflection points and the curve sketching checklist
Worked example · free
Absolute extrema of a quartic on a closed interval
- 2g is a polynomial, so it is continuous on [−1, 3] and the Extreme Value Theorem guarantees both extrema exist. Differentiate: g′(x) = 4x³ − 16x = 4x(x − 2)(x + 2).
- 1Critical points are x = 0, 2, −2; only x = 0 and x = 2 lie in [−1, 3]. g′ exists everywhere, so no other candidates arise.
- 2Evaluate at critical points and endpoints: g(−1) = 1 − 8 + 3 = −4, g(0) = 3, g(2) = 16 − 32 + 3 = −13, g(3) = 81 − 72 + 3 = 12.
- 1Compare the four values: the largest is 12 and the smallest is −13.
Key terms
- Critical Point
- A critical point is an input c in the domain where f′(c) = 0 or where f′(c) does not exist. Every interior local extremum of a function occurs at such a point, but not every such point is an extremum.
- Absolute Maximum
- An absolute maximum is a value f(c) with f(x) ≤ f(c) for every x in the domain. On a closed bounded interval it is found by comparing f at all critical points and at both endpoints.
- Indeterminate Form
- An indeterminate form is a limit shape, such as 0/0 or ∞/∞, whose value cannot be decided from the limits of its parts alone. Different functions with the same shape can give any answer.
- Inflection Point
- An inflection point is a point on a graph where concavity changes between up and down. For twice differentiable functions, f″ must equal zero there, though a zero of f″ alone does not guarantee a switch.
- Concave Up
- Concave up describes a graph near a point where the derivative f′ is increasing, so the curve bends upward like a cup. When f″ exists, positive values of f″ indicate this shape.
Extreme Values, Indeterminate Forms and Curve Sketching FAQ
Can I use L'Hôpital's Rule on any limit that looks hard?
No. The rule only applies when the quotient is of type 0/0 or ∞/∞, and f and g must be differentiable near the point. Applying it to a determinate quotient such as 0/3 produces wrong answers, so substitute first and confirm the type on every line.
What do I do when the second derivative test gives zero?
A zero second derivative at a critical point means that test gives no verdict. Either switch to a sign chart of f′ around the point, or find the first higher derivative that is nonzero there: odd order rules out an extremum, even order gives a max or min by its sign.
Do I need to check endpoints when finding a global maximum?
Yes. On a closed interval the largest value often occurs at an endpoint, where the derivative need not vanish. List both endpoints alongside every critical point, evaluate the function at each, and choose the biggest and smallest outputs from that list.
How do I evaluate a limit like x ln x as x goes to zero?
That product is type 0 · ∞, so rewrite it as a quotient before differentiating. Placing x in the denominator as 1/x gives ln x divided by 1/x, of type ∞/∞, and one application of the rule produces −x, which tends to zero.
Exam move
Build one sign chart by hand each day for a factorable derivative until it takes under three minutes. For L'Hôpital practice, keep a two-column log: the left column records the form (0/0, ∞/∞, or a number) on each line, the right column the derivative step. Redo any problem with a blank left entry. End each week by sketching two curves using all seven checklist items without a graphing tool.
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