Uses of Derivative in Mathematics & EconomicsIncreasing and Decreasing FunctionsConcavity and ConvexityRelative Extrema

Increasing and Decreasing Functions

Definition of Increasing and Decreasing Function

If a graph is going up is that its slope is positive. If the graph is going down, then the slope will be negative.  Since slope and derivative are synonymous, we can relate increasing and decreasing with the derivative of a function.

Definition of Increasing and Decreasing Function A function is increasing on an interval if

for any x1 and x2 in the interval then x1 <  x2          implies          f(x1)  <  f(x2) A function is decreasing on an interval if

for any x1 and x2 in the interval then x1 <  x2          implies          f(x1)  >  f(x2)

Theorem on Derivatives and Increasing/Decreasing Functions Let f be a differentiable function on the

interval (a,b) then  If f '(x) < 0 for x in (a,b), then f is

decreasing there.

If f '(x) > 0 for x in (a,b), then f is increasing there.

If f '(x) = 0 for x in (a,b), then f is constant.

Concavity and Convexity

General definitions Let f be a function of a single variable

defined on an interval. Then f is concave if every line segment joining two points on its graph is never above the graph

convex if every line segment joining two points on its graph is never below the graph.

General definitions

Relative Extrema

Definition of an Extrema The extrema of a function f are the values

where f is either a maximum or a minimum. Let f  be a function defined on the

interval (a,b) containing the point c. Then f  has a minimum at c if f (c) <  f (x) for

all x in (a,b).

f  has a maximum at c if f (c) >  f (x) for all x in (a,b).

Definition of a Relative Extrema Let f  be a function defined on the

interval (a,b) containing the point c.  Then f  has a relative maximum at c if  

          f (c) >  f (x) for all x in some interval (u,v) containing c.

f  has a relative minimum at c if            f (c) <  f (x) for all x in some interval (u,v) containing c.

Figure

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3.5Relative Max at x=2

Figure

0.5 1 1.5 2 2.5 3 3.50

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1.52

2.53

3.5Relative Minimum at x=2

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