Relationship between Intermediate Theorem and Mean Value Theorem

The intermediate value theorem is a continuous function theorem that deals with continuous functions. The intermediate value theorem is important in mathematics, and it is particularly important in functional analysis. This theorem illustrates the advantages of a function’s continuity in more detail. In mathematics, the two most important examples of this theorem are frequently employed in many applications. 

Intermediate Value Theorem statement: 

Suppose “f” is a continuous function over the closed interval [a, b], and its domain contains the values f(a) and f(b) at the interval’s ends, then the function takes any value between the values f(a) and f(b) at any point within the interval, according to the intermediate value theorem. This theorem is explained in two ways: first, as follows: 

Statement 1: 

If k is a number that falls between f(a) and f(b), then

In either case, f(a) < k < f(b) or f(a) > k > f(b) is true,

There exists at least one number c between a and b, that is, c ∈ (a,b) in a way that the function f(c) = k is satisfied. 

Statement 2: 

Set of images of a function in the interval [a, b], containing [f(a), f(b)] or [f(b), f(a)], that is, 

either f([a, b]) ⊇ [f(a), f(b)] or f([a, b]) ⊇ [f(b), f(a)] 

Theorem explanation: 

The statement of the intermediate value theorem appears to be difficult to comprehend. However, it is possible to understand it in simpler terms. Assuming the above picture has a continuous function f with endpoints a and b, the heights of the points “a” and “b” would be denoted by the letters “f(a)” and “f(b), respectively.

If we choose a height k between these heights f(a) and f(b), then according to this theory, this line must cross the function f at some point (say c), and this point must be located between the heights a and b, as shown in the diagram.

Bolzano’s theorem is an intermediate value theorem that holds if c = 0. It is also known as Bolzano’s theorem. 

Intermediate Theorem examples: 

Question: 

Verify that the equation x⁵–2x³–2=0 has a solution in the interval [0-2], or whether there is no solution at all. 

Solution: 

Let’s see if we can discover the values of the above function at the points x = 0 and x = 2 in the graph. 

f(x) = x⁵ – 2x³ -2 = 0 

In the given function, substitute x = 0 for the value of x. 

f(0) = (0)⁵ – 2(0)³ -2

f(0) = -2 

In the given function, substitute x = 2 for the value of x. 

f(2) = (2)⁵ – 2(2)³ -2

f(2) = 32 – 16 – 2

f(2) = 14  

Because of this, we may state categorically that the curve is below zero at x = 0, and that it is above zero at x = 2.

Due to the fact that the provided equation is a polynomial, the graph of the equation will be continuous.

According to the intermediate value theorem, we can conclude that the graph must cross at some point between the two points (0, 2).

Thus, there exists a solution to the equation x⁵ + 2x³ -2 = 0 in the interval [0, 2], which is between the numbers 0 and 2. 

Mean Value Theorem: 

Among the most valuable tools in both differential and integral calculus, the mean value theorem is one of the most important. It has extremely important ramifications in differential calculus, and it aids us in understanding the identical behaviour of different functions when comparing them.

Several similarities may be found between the hypothesis and conclusion of the mean value theorem and those of The Intermediate Value Theorem. The Lagrange’s Mean Value Theorem is another name for the mean value theorem. The abbreviation for this theorem is MVT. 

Mean Value Theorem statement: 

Assume that f(x) is a function that meets the requirements listed below: 

• f(x) is Continuous in [a,b]

• f(x) is Differentiable in (a,b) 

Then there is a number c, s.t. a < c < b, 

f(b) – f(a) = f ‘(c) (b – a) 

Proof of Mean Value Theorem: 

If we take into account the function h(x) = f(x) – g(x), where g(x) is the function representing the secant line AB, the mean value theorem can be proven to be true. Rolle’s theorem may be extended to the continuous function h(x), and it has been demonstrated that there exists a point c in the coordinates (a, b) where h’(c) = 0. The conclusion of the mean value theorem will be reached as a result of this equation.

Consider a line going through the points (a, f(a)) and (b, f(b)) and then continuing on. The line’s equation is 

y – f(a) = {f(b) – f(a)}/(b-a) . (x – a)

Or y = f(a)+ {f(b) – f(a)}/(b-a) . (x – a) 

Assume h is a function that defines the difference between any function f and the line preceding it. 

h(x) = f(x) – f(a) – {f(b)-f(a)}/(b-a) . (x – a) 

We have the result of “Rolle’s theorem” by utilising 

h’(x) = f’(x) – {f(b)-f(a)}/(b-a)

Or f(b) – f(a) = f’(x) (b – a). 

Hence Proved.

Physical interpretation of Mean Value Theorem: 

Because (f(b)-f(c))/(b-a) is the average change in the function across [a, b], and f’(c) is the instantaneous change at ‘c,’ the mean value theorem asserts that the instantaneous change at some interior point is equal to the average change in the function over the interval. 

Conclusion: 

The mean value theorem is not the same as the intermediate value theorem, and the two are not equivalent. It is the differentiable functions and derivatives that are the focus of the mean value theorem, whereas the continuous functions and derivatives are the focus of the intermediate theorem. The mean value theorem ensures that the derivatives have certain values, whereas the intermediate value theorem ensures that the function has certain values between two specified values, as opposed to the derivatives.