concave up/concave down The graph of a function
y= f(x) may be described as concave up over an inter-
val if, over that interval, the slope of the tangent line to
the curve increases as one moves from left to right.
Assuming the function is twice differentiable, this
means that the
DERIVATIVE
f′(x) is a strictly increasing
function and, consequently, the double derivative satis-
fies f′′(x)>0. (See
INCREASING
/
DECREASING
.) For exam-
ple, the double derivative of f(x) = x2is always positive,
f′′(x) = 2>0, and the parabola y= x2is concave up. A
concave-up graph also has the property that any
CHORD
joining two points on the graph lies entirely
above the graph.
The graph of a function y= f(x) is concave down
over an interval if, over that interval, the slope of the
tangent line to the curve decreases as one moves from
left to right. Assuming the function is twice differen-
tiable, this means that f′(x) is a strictly decreasing func-
tion and, consequently, f′′(x)<0 for all points on the
interval. As an example, since the double derivative of
f(x) = x3is negative only for negative values of x(f′′(x)
= 6x<0 for x<0), we have that the cubic curve y= x3is
concave down only to the left of the y-axis. A concave-
down graph has the property that any chord joining
two points on the graph lies entirely below the graph.
A point at which the concavity of the graph
changes is called an inflection point or a point of inflec-
tion. (Alternative spelling: inflexion.) If x= ais a point
of inflection for a twice-differentiable curve f(x),then
the double derivative f′′(x) is positive to one side of
x=aand negative to the other side. It must be the case
then that f′′(a) = 0. The converse need not hold, how-
ever. The function f(x) = x4, for example, satisfies
f′′(0) = 0, but the concavity of the curve does not
change at x= 0.
A study of the concavity of a graph can help one
locate and classify local maxima and minima for
the curve.
See also
GRAPH OF A FUNCTION
;
MAXIMUM
/
MINIMUM
.
concentric/eccentric Two circles or two spheres are
called concentric if they have the same center. Two fig-
ures that are not concentric are called eccentric. The
region between two concentric circles is called an
ANNULUS
.
concurrent Any number of lines are said to be con-
current if they all pass through a common point. Many
interesting lines constructed from triangles are concur-
rent. Two lines a1x+ b1y= c1and a2x+ b2y= c2in the
Cartesian plane are concurrent if a1b2– a2b1 ≠ 0.
See also
TRIANGLE
.
conditional (hypothetical) In
FORMAL LOGIC
a state-
ment of the form “If … then…” is known as a condi-
tional or an implication. For example, “If a polygon
has three sides, then it is a triangle” is a conditional
statement.
A conditional statement has two components: If p,
then q. Statement pis called the antecedent (hypothesis,
or premise) and statement qthe consequent (or conclu-
sion). A conditional statement can be written a number
of different, but equivalent, ways:
If p, then q.
pimplies q.
qif p.
ponly if q.
pis sufficient for q.
qis necessary for p.
It is denoted in symbols by: p→q.
conditional 89
Concavity