Learning Maximum and Minimum Values Theorems

“Minimum and maximum value” states local function extrema where the purpose has a total maximum value at “x = c if f(c) ≥ f (x)”. Consider for every value x ∈ Domain (f). Extreme value theorem for learning minimum and maximum value rule or theorem is used to prove “Rolle’s theorem”. The backbone of the technique is demonstrated as a theorem which is defined as “Fermat’s stationary point” or “Fermat’s theorem”. Derivatives in the theorem are regarded as an essential concept that emerges enormously helpful in different applications. Proofs from derivative applications relate to the function of obtaining maximum and minimum value theorems. 

Extreme value theorem: Concept

• The application of the extreme value theorem is crucially important in demonstrating the minimum and maximum function value on a particular interval. 
• It is assured by the extreme value theorem that ramification of “maximum and minimum value” under some specific situations. 
• The extreme value theorem is demonstrated when a particular point is referred to as a “minimum point” if the function value at that point is considered less as compared to “function values” for every “x value” in the certain interval phase. 

Consider the instance of finding the EVT of “f(x) = 2×3 – 3×2 on the interval [0, 3]” which includes the four stages. The first stage included is finding the crucial numbers of f (x) over (a, b), the open interval. The second stage is considered as the evaluation process of f (x) at every crucial number. The third stage is referred to as the analysis of f (x) at every endpoint over [a, b], the closed interval. The last stage is associated with obtaining the minimum values which are “minimum” and the largest is “maximum”.

Fermat’s theorem: overview

Fermat’s theorem states that for any prime number “p” and any sort of integer “a” such that the prime number does not dissect the integer and “p” divides exactly into ap − a. However, the number “n” is not considered in dividing an – a, where some “a” is mentioned as the composite number. For instance, let “a” is equal to 2 and n is equal to 341, then “a” and “n” are considered prime and 341 divides into 2341 – 2. Hence, it is obtained that “341 = 11* 31”, so it is regarded as a composite number. Thus Fermat’s theorem provides that the test is important but not adequate primarily. It is mentioned in most of the books that Fermat’s own derivation of proof does not exist.

Proofs of Derivative application: Facts

Derivatives in mathematics help to say individual regarding what velocity a person is driving or assist in assuming the stock market fluctuations. The proof of application of derivatives is considered from a different theorem. In this context, Fermat’s theorem is considered where f (X) has concerned extrema at x is equal to c and f’ (c) still exists then x is equal to c which is demonstrated as the crucial point of x. In this fact, it is the crucial point where f’ (c) is mentioned as zero. It is assumed that f (x) has a concerned minimum to do the nearly identical proof and the relative maximum at x is equal to c. The proof has been done with extreme value theorem whose results reflect f’ “(c) ≤ 0 and f’ (c) ≥ 0 and x= c”

Conclusion

The extreme value theorem has critically proved the existence of worldwide minima and maxima for constant function in the interim. EVT is regarded as intuitive and looks similar to a primary property of a constant function. In mathematics, the value theorem is demonstrated as highly important as its consequences in real analysis. The importance of Fermat’s theorem is demonstrated in the study which assists in calculating integer’s power modulo “prime numbers.” Basically, the calculation of length and breadth becomes easy with Fermat’s Theorem. The application of derivatives in the study is regarded as highly important which demonstrates the “change rate of a quantity”, “decreasing and increasing functions”, and “normal and tangent to a curve”.