A cubic function is a polynomial of degree 3 ; that is, it has the form f(x)=a x^3+b x^2+c x+d,

A cubic function is a polynomial of degree 3 ; that is, it...

Key Concepts

Local Extreme Values

Local extreme values refer to the local maximums and minimums of a function, which occur at critical points where the derivative is zero and changes sign. For cubic functions, these are the possible peaks and valleys in the graph, and since the derivative is quadratic, a cubic function can have up to two local extreme values.

Cubic Function

A cubic function is a type of polynomial function of degree three, typically written in the form f(x) = ax³ + bx² + cx + d with a ? 0. This function is fundamental in algebra and calculus, exhibiting behaviors such as inflection points and potentially multiple turning points due to its odd degree.

Polynomial Degree

The degree of a polynomial, in this case three for a cubic function, indicates the highest power of x present in the function. This degree dictates key characteristics such as the maximum number of turning points and the end behavior of the function, which are central in understanding the graph’s overall structure.

Derivative

The derivative of a function measures its instantaneous rate of change and is instrumental in identifying where the function’s slope is zero or undefined. For a cubic function, taking the derivative leads to a quadratic function that is critical for pinpointing potential local maxima and minima.

Critical Numbers

Critical numbers are the values of x where the derivative of a function is zero or does not exist. In the context of cubic functions, setting the quadratic derivative equal to zero allows us to find the critical numbers, which help in determining the locations of local extreme values or inflection points.

Discriminant Analysis

When the derivative of a cubic function is a quadratic equation, the discriminant (the part under the square root in the quadratic formula) determines the number of real roots. A positive discriminant yields two distinct critical numbers, a zero discriminant yields one repeated (double) critical number, and a negative discriminant means there are no real critical numbers.

Transcript

00:01 A cubic function is a polynomial of degree 3 of the form f of x equal axq plus bx squared plus cx plus d, where a is different from zero.

00:18 In part a we will show that a cubic function can have 2, 1 or no critical numbers.

00:23 And we give examples and sketches to the straight these possibilities.

00:28 Part b, we discuss about how many local extreme values.

00:32 Can a cubic function have.

00:35 So in part a, we're going to start by remembering the definition of critical point.

00:42 And we know that c on the domain of the function is a critical point of f or a critical number if the derivative at that point does not exist or it exists and is equal to so this is the definition of critical number we are going to use.

01:29 And so first of all, we know that the domain of a cubic function is the real numbers, the set of real numbers.

01:42 So if f is a cubic function, then the domain of f is the real numbers.

01:56 That is the function is defined at every real number.

02:05 That is because the polynomial is defined everywhere for any real number x.

02:10 We have this and then the derivative of f is given by, we have to find the derivative of this general expression here, is 3ax squared plus 2bx plus c.

02:33 And we know from the statement of the problem that a is not zero.

02:41 If a is not zero, sorry here i made a mistake, is 3ax squared.

02:50 So because a is not zero, this term is not zero, and then it's a polynomial of degree 2.

03:00 But before saying that, we can see that this derivative is defined everywhere.

03:05 That is for any real number x.

03:08 It means that this possibility is not an option in this case.

03:16 So we can say that f derivative is defined at every real number x.

03:37 And the conclusion of that is that the critical numbers of this function, this cubic function, are those real values.

03:47 X where the derivative is equal to zero.

03:53 The only critical numbers are those values x in the real numbers such that the derivative of f and x is equal to zero.

04:17 That is because this option in the definition can happen in the case of the cubic function because the derivative is also a polynomial in this case of degree 2 and is defined everywhere in the real numbers.

04:34 So the only possibility for a value x to be a critical number of cubic functions is that the derivative at that point is equal to 0.

04:46 And it could be any principle can be any real number because the domain of the cubic function is the set of real numbers.

04:58 So we've got to find this equation here.

05:02 But the equation f derivative equal to zero is the same as it's equivalent to to the equation of 3a x squared plus 2bx plus c equals 0.

05:28 And we know that with a different from 0, this equation has 2, 1, or 0 roots.

05:39 Then can have 2, 1, or no roots.

06:01 Okay, so if we take into account what we said before, it means that the function f can have 2, 1, or no critical numbers.

06:30 Here i put this way because we can have 1, so in that case, will be no root or no critical number.

06:38 So this is the situation.

06:43 We have a cubic function.

06:45 That cubic function is defined everywhere in the real numbers, and its derivative is also defined everywhere in the real numbers.

06:52 And so the only possibilities for critical numbers for this function are the real values of x for which the derivative, the first derivative is zero.

07:03 And this becomes second degree, equation which we know can have two one or no roots and then this implies that the function can have only two one or no critical now so let's put some examples of that so i'm gonna take first the function x q minus 3x square plus 2x, which can be factored out as x times x minus 1 times x minus 2.

07:53 It can be verified easily, that this is a factorization of this function...

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