Investigation of the function. Extremum of the function. Necessary and sufficient conditions for the existence of an extremum. Convexity, concavity and inflection points. Asymptote. The overall study of design features

1. Monotonic conditions. Extremum of function

2. Convexity and concavity. Point of inflection

3. Àsymptotes

4. General Scheme for the Investigation of the Graph of a Function

Monotonic conditions. Extremum of function

Theorem 1. Let be continuous on [a, b] and differentiable on (a, b). is a constant

function if and only if for all .

Definition 1. A function is said to be monotonic increasing (resp. monotonic decreasing )

or simply increasing ( resp. decreasing ) on an interval if and only if , if

then (resp. , if , then ).

Definition 2. A function is said to be strictly increasing ( resp. strictly decreasing ) on an

interval if and only if , if then (resp. , if ,

then ).

Theorem 2. Let be continuous on [a, b] and differentiable on (a, b). Then

(a) if is strictly increasing on [a, b]; and

(b)if is strictly decreasing on [a, b].

Definition 3. A neighborhood of a point is an open interval containing , i.e. is a neighborhood of for some .

Definition 4. A function is said to attain a relative maximum (minimum ) at a point if

( ) in a certain neighborhood of , i.e. such that ( ) for .

Theorem (Fermat Theorem).

Given is a point defined on and differentiable at a point if has an extreme value ( max. or min ) , then .

Note has maximum or minimum at .

Definition 5. (a) A turning point is a maximum or minimum point.

(b) If , then is called a critical or

stationary value and its corresponding point on the graph is called stationary point.

Notes

1. turning point stationary point

2. turning point + differentiable stationary point

3. stationary point turning point

Convexity and concavity. Point of inflection

Let be a function differentiable on an interval J. The function f is called convex (concave) on J, if all points of its graph on J lie above (below) any tangent line to on this interval (excepting point of tangency). Let f be continuous at a point . If there exists such that f is concave (convex) in and convex (concave) in , the point is called the point of inflection of .

The second Derivative Test for Concavity and Convexity:

If , for each , then is convex on J,

if , for each , then is concave on J.

It follows:

If is continuous at and ( ) in and ( ) in , then is a point of inflection.

Moreover: If is a point of inflection of f, then either or doesn’t exist.

If f is three times differentiable at a point , and , then is a point of inflection.

Definition 6. Given that is continuous on , if any such that

(i) (ii)

Concave Downward Concave Upward

Theorem 3. If is a function on such that is second differentiable on then

(i) iff is concave upward on

(ii) iff is concave downward on .

Definition 7. Let be a continuous function. A point on the graph of is a point of inflexion (point of inflection) if the graph on one side of this point is concave downward and concave upward on the other side. That is, the graph changes concavity at .

Note A point of inflexion of a curve must be a continuous point but need not be differentiable there. In Figure (c), R is a point of inflexion of the curve but the function is not differentiable at .

Theorem 4. If is second differentiable function and attains a point of inflexion at , then

.

Note:

(i) max. or min. point but not derivative.

(ii) point of inflexion may not be obtained by solving where and such that .

(iii) Let be a function which is second differentiable in a neighborhood of a point of inflexion iff does not change sign as increases through (sign gradient test)

– if and , then attains a relative max. or relative min.

– if and , then attains an inflexion point at .

Asymptotes

If a function f gets close to a certain number L when x gets larger and larger, then we say that the limit as x goes to infinity is L and we write: . Likewise, if f gets close to L when x gets smaller and smaller, then the limit as x goes to negative infinity is L and we write: . In both cases, the line y = L is a horizontal asymptote of f.

Example: because when x is very large, is close to 0. The x-axis is a horizontal asymptote of the function :

If a function gets larger and larger as x gets close to a number a, then it “goes to infinity” and we write: . The line x = a is a vertical asymptote of f. Similarly, if gets smaller and smaller as x gets close to a, then it “goes to negative infinity” and we write: . Again, the line x = a is a vertical asymptote of f.

An important result:

If then . This is because one over a very large positive number and one over a huge negative number are both close to 0.

Inclined asymptotes have an equation y = kx + q and their position is arbitrary except vertical. In order a straight line y = kx + q be an asymptote, the coefficients k and q must satisfy at least one pair of the following conditions

and (k and b are numbers).

Naturally these limits must be finite real numbers. A certain function can have maximally two inclined asymptotes.

General Scheme for the Investigation

of the Graph of a Function

The following information is useful for sketching the graph of

(1) The domain of , i.e. the range of values of within which is well-defined.

(2) Determine whether is periodic, odd or even, so that the graph may be symmetric about the coordinate axes or about the origin.

(3) Turning points and monotonicity of .

(4) Inflectional points and convexity of .

(5) Asymptotes including horizontal, vertical and oblique ones (if any).

(6) Some special points on the graph, such as intercepts.