Ex 5.1.16 Show that a cubic polynomial can have at most two critical points. Give examples to show that a cubic polynomial can have zero, one, or two critical points. Give examples to show that a cubic polynomial can have zero, one, or two critical points. 5.1 Extreme Value of Functions 5.2 Mean Value Theorem 5.3 Connecting f’ and f’’ with the Graph of f BC Calculus Critical Point: a point in the interior of the domain of a function f at which f ′=0 or f ′ does not exist is a critical point of f. 5.1 Critical points; 5.2 Intervals of increase and decrease; 5.3 Points of inflection; 5.4 Intervals of concave up and down; 6 Symmetry; 7 Integration. 7.1 First antiderivative. 7.1.1 Direct computation; 7.1.2 Computation in terms of functional equations for the logistic function.

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Critical Points of Functions of Two Variables. A critical point of a multivariable function is a point where the partial derivatives of first order of this function are equal to zero. Examples with detailed solution on how to find the critical points of a function with two variables are presented.

• 1Definition
• 3Differentiation
  • 3.2Second derivative
• 4Functional equations
• 5Points and intervals of interest
• 7Integration
  • 7.1First antiderivative

Definition

The logistic function is a function with domain and range the open interval , defined as:

Equivalently, it can be written as:

Yet another form that is sometimes used, because it makes some aspects of the symmetry more evident, is:

For this page, we will denote the function by the letter .

We may extend the logistic function to a function , where and .

Probabilistic interpretation

The logistic function transforms the logarithm of the odds to the actual probability. Explicitly, given a probability (strictly between 0 and 1) of an event occurring, the odds in favor of are given as:

This could take any value in

The logarithm of odds is the expression:

If equals the above expression, then the function describing in terms of is the logistic function.

Key data

Differentiation

First derivative

Consider the expression for :

We can differentiate this using the chain rule for differentiation (the inner function being and the outer function being the reciprocal function . We get:

Simplifying, we get:

We can write this in an alternate way that is sometimes more useful. We split the expression as a product:

The first factor on the right is , and the second factor is , so this simplifies to:

Second derivative

Using the expression for

From the above, we have:

Differentiating both sides, we obtain:

This simplifies to:

We can now re-use the expression for and obtain:

Using the expression for

We have:

Using the product rule for differentiation and the chain rule for differentiation, we get:

Note from the expression that shows that is even, so we can rewrite as , and we get:

We can re-use the expression and obtain:

Functional equations

Symmetry equation

The logistic function has the property that its graph has symmetry about the point . Explicitly, it satisfies the functional equation:

We can see this algebraically:

Multiply numerator and denominator by , and get:

Differential equation

As discussed in the #First derivative section, the logistic function satisfies the condition:

Therefore, is a solution to the autonomous differential equation:

The general solution to that equation is the function where . The initial condition at pinpoints the logistic function uniquely.

Points and intervals of interest

Critical points

The function has no critical points. To see this, note that the derivative is:

Note that the numerator is never zero, nor is the denominator. Therefore, the function is always defined and nonzero.

Intervals of increase and decrease

The derivative:

is always positive. So the function is increasing on all of .

The asymptotic values are:

and:

In other words, the range of the function is the open interval , and it increases throughout its domain.

Points of inflection

The second derivative is:

We already noted that is always defined and nonzero, so the only way for to be zero is if < or . This solves to:

This solves to , or .

Thus, the second derivative is 0 at the point , i.e., with and .

Intervals of concave up and down

As above, we have:

We also noted that for all . Therefore, for and for . Therefore, is:

• concave up for , i.e.,
• concave down for , i.e.,

Symmetry

We discussed above a functional equation satisfied by :

From this, the following can be deduced:

• The graph of has half-turn symmetry about the point .
• is an even function. Note that this can also be seen from the actual expression: . But we don't need the actual expression to deduce that it is even -- the functional equation above gives evenness.
• is an odd function. This can be directly deduced from being even, but can also be verified from the actual expression: .

Integration

First antiderivative

Direct computation

We have:

Computation in terms of functional equations for the logistic function

We have:

We also have that , so we get:

This can be rewritten as:

By the chain rule for differentiation, we get:

Thus:

This can be simplified and verified to be the same as the answer obtained by direct computation.

Calculus requires knowledge of other math disciplines. To make studying and working out problems in calculus easier, make sure you know basic formulas for geometry, trigonometry, integral calculus, and differential calculus.

20 Handy Derivatives for Calculus

These derivatives are helpful for finding things like velocity, acceleration, and the slope of a curve — and for finding maximum and minimum values (optimization) when you’re dealing with differential calculus.

Product rule:

Quotient rule:

List of 20 handy derivatives:

Note: Where c is a constant

Note: Where c is a constant

Note: Where a is a constant

20 Handy Integrals for Calculus

If you’re studying integral calculus, the following integrals will help you to work out complex calculations involving area, volume, arc length, center of mass, work, and pressure.

Note: Where a is a constant

5.1 Critical Valuesap Calculus Transcendentals

Note: Where a is a constant

Note: Where a is a constant)

Note: Where a is a constant

Note: Where a is a constant

Handy Trigonometry: Right Triangles, Degrees and Radians, Identities, and Formulas

This trigonometry info will help you deal with triangles, finding their relationships between the sides and angles of right triangles, and make calculations based on these relationships.

Right triangle trig

SohCahToa:

Degrees and radians

5.1 Critical Valuesap Calculus Solver

Identities

Reciprocal identities

Quotient identities:

5.1 Critical Valuesap Calculus 2nd Edition

Pythagorean identities:

Formulas

5.1 Critical Valuesap Calculus 14th Edition

Half-angle formulas:

Double-angle formulas:

Reduction formulas: