Theory

Curve sketching

The complete scheme of the study, the criteria for maxima, minima and inflection points, convexity and the construction of the qualitative graph.

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The scheme of the study
Monotonicity and stationary points

The scheme of the study

Scheme of curve sketching

Studying a function means deriving from its expression the properties that determine its graph. Each step rests on a result of the theory:

domain: conditions for the expression to be defined (nonzero denominators, positive arguments of logarithms, nonnegative radicands of even-index roots), to be imposed as a system when there is more than one;
symmetries and periodicity: $f (- x) = f (x)$ for an even function (graph symmetric with respect to the $y$ -axis), $f (- x) = - f (x)$ for an odd one (graph symmetric with respect to the origin), $f (x + T) = f (x)$ for a periodic one, which is studied on a single period;
sign and intersections with the axes: the inequality $f (x) > 0$ and the equation $f (x) = 0$ ; when the equation cannot be solved algebraically, the zeros are located with the intermediate zero theorem;
limits at the endpoints of the domain and asymptotes: definitions of limit and of asymptote from the chapter on limits;
continuity: theorem on the continuity of elementary functions and of operations between continuous functions;
differentiability and first derivative: differentiation rules from the chapter on derivatives; at junction points, limit of the difference quotient or one-sided limits of $f^{'}$ ;
monotonicity and stationary points: monotonicity test;
relative and absolute maxima and minima: Fermat’s theorem, first derivative test, second derivative test, Weierstrass’s theorem;
concavity, convexity and inflection points: characterization of convexity through the second derivative;
qualitative graph: synthesis of all the information in a consistent drawing.

Remark

The steps check one another. If the sign study gives $f (x) > 0$ on an interval and the computation of limits gives $f (x) \to - \infty$ at one of its endpoints, at least one of the two results is wrong; if the function is even, maxima, minima and inflection points appear in symmetric pairs. A consistency check between the steps detects computational errors before the graph is drawn.

Monotonicity and stationary points

1. Monotonicity test

Let $f$ be continuous on an interval $I$ and differentiable at the interior points of $I$ .

If $f^{'} (x) > 0$ at every interior point, $f$ is strictly increasing on $I$ .
If $f^{'} (x) < 0$ at every interior point, $f$ is strictly decreasing on $I$ .
If $f^{'} (x) = 0$ at every interior point, $f$ is constant on $I$ .

The theorem follows from Lagrange’s mean value theorem and the proof is in the chapter on derivatives. In curve sketching one solves the inequality $f^{'} (x) > 0$ and records the result in the sign chart: on each interval where $f^{'}$ has constant sign the monotonicity is determined.

2. Stationary point

A point $x_{0}$ interior to the domain, with $f$ differentiable at $x_{0}$ and $f^{'} (x_{0}) = 0$ , is called stationary: the tangent line at $(x_{0}, f (x_{0}))$ is horizontal. Stationary points are the natural candidates for interior relative maxima and minima, and their nature is established with the first and second derivative tests.

Warning

The monotonicity test holds on an interval, not on a union of intervals. For $f (x) = \frac{1}{x}$ we have $f^{'} (x) = - \frac{1}{x ^{2}} < 0$ at every point of the domain, therefore $f$ is decreasing on $(- \infty, 0)$ and on $(0, + \infty)$ separately; on the union it is not, because $f (- 1) = - 1 < 1 = f (1)$ .

y = \dfrac{1}{x}

y = \frac{1}{x}

decreases on

(- \infty, 0)

and on

(0, + \infty)

, but

f (- 1) = - 1 < 1 = f (1)

The theory continues with 6 more sections (relative maxima and minima, tests for the nature of stationary points, absolute maxima and minima…), plus all worked examples, the solutions and the PDF.

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