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Intermediate Value Theorem

Referencing Styles : APA | Pages : 1

Intermediate Value Theorem

The intermediate value theorem describes a key function of continuous function in the field of Mathematics. The intermediate value theorem states that if a continuous function has two values then the same would be a function whose graph can be drawn without lifting the pencil from the paper. In a mo0re detailed version, it can be stated that if a function f with an interval [a, b] as its domain takes respective values of f(a) and f(b) at each of the interval then the same also takes value between f(a) and f(b) at some point of time. The two major characteristic which is associated with the intermediate value theorem are discussed below in point form:

  1. If a continuous function depicts a value which is of the opposite sign inside an interval then the same has a root in the interval. This is stated by Bolzano’s Theorem.
  2. The image of the continuous function over an interval is itself an interval in relation to intermediate value function.

The intermediate value theorem states the intuitive property of the continuous functions and on the basis of the same values are efficiently represented in the function.

 

A graph is presented above which depicts the presentation of Intermediate Value Theorem and effectively shows the functions of f(a) and f(b). The graph shows that f(x0) forms the middle point between the two functions which proves that the theorem has applications. The above graph shows that y = f(x) and the variables which are used for this function are continuous in nature making the same suitable for the application of Intermediate Value Theorem. In a nutshell, it can be said that Intermediate Value Theorem, where there are two real numbers a and b where a < b. let f be continuous function between the interval of [a, b]. Then for every y0​ between f(a) and f(b), there exists a number x0 in ​∈ [a, b] with f(x0​) = y0​.

Application of Intermediate Value Theorem with Examples

The most important application which can be identified for the intermediate value theorem is that it can effectively prove that some of the equations would have a solution and has a mid-point in terms of functions which is represented in the equation. The usefulness of the intermediate value function in terms of its application is that there is a point in some of the line and there is a point which is below the line and the curve is continuous in nature.

An example which can be provided in respect of an equation which is x5 – 2x3 – 2 = 0 and the interval which is provided is stated as x = 0 and x = 2. The application of the theorem requires first to prove that the equation is continuous in nature and the same can be shown by substituting the value of the x in the equation in a systematic manner.

At x = 0,

x5 – 2x3 – 2

= 05 – 2 × 03 – 2

= -2

At x = 2,

x5 – 2x3 – 2

= 25 – 2 × 23 - 2

= 32 – 16 – 2

= 14

The above analysis shows that when the value of x = 0 that the curve would lie below zero and accordingly demonstrated in a graph. On the other hand, when the value of x = 2 that the curve would lie above zero and similarly demonstrated in the graph. It is further to be noted that the equation is polynomial in nature and therefore the curve would also be continuous and so somewhere in between the curve must cross through y = 0. The analysis effectively shows that there exists a possible solution between the equation x5 – 2x3 – 2 = 0 in the interval [0, 2].

Another example which can be provided in respect of intermediate value theorem is in case of a function f which is continuous on [0, 1] and f (0) = f (1). Now let n be the positive and then it must be proved that there is some number x such that

                                                                  f(x)=f(x+n1​)

The viable solution in respect of the above equation can be provided in a manner which is shown below in details:

Define g(x) =

Consider the set of numbers S =

Finally, if the largest number in S is f(0) = f(1)f(0)=f(1), then the same argument works with kk chosen such that f\left(\frac{k}{n}\right)f(nk​) is the minimum number in SS. _\square□​

Note that if f(0) is both the largest and smallest number in SS, then they are all the same and 

Formulas of Polynomial Functions

The formula of a polynomial function can be defined on the basis of the equation which is formed. A general function is defined to be polynomial if the same is in the form of f (x) = anxn + an-1xn-1 + ... a2x2 + a1xa0 where the values of an, an-1….a1 and a0 are real numbers while n stands for the nonnegative integer and an ≠ 0. This can be appropriately presented in a graph or an equation format. The polynomial functions which are used in algebra which can be solved effectively by following factoring. If a polynomial of lowest degree p has zeros at x=x1,x2,…,xnx=x1,x2,…,xn, then the polynomial can be written in the factored form.

f(x)=a(x−x1) p1 (x−x2)p2⋯(x−xn)pn . The equation considers the power factor on each of the constants which is represented above. The powers pi on the equation can be determined from the behaviour of the graph at the corresponding intercept and the stretch factor a can be determined given a value of the function other than the x-intercept. The formulas of the polynomial functions are considered to be important for solving important equations in algebra and also present the same in graph format.

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