Extreme Value Theorem
if a function is continuous on a closed interval [a,b], then the function must have a maximum and a minimum on the interval. Used to determine if a function has a minimum or a maximum on the interval.
The Extreme Value Theorem states that a continuous function on a closed interval [a, b] must have both a maximum and a minimum value on that interval.
This means that if f(x) is a continuous function on the interval [a, b], there must exist values c and d in [a, b] such that f(c) is the maximum value of f(x) on the interval [a, b], and f(d) is the minimum value of f(x) on the same interval.
In other words, no matter how wildly a continuous function may oscillate or curve on a closed interval, it must attain its highest and lowest points somewhere on that interval.
This theorem is widely used in calculus and mathematical modeling to find optimal solutions and to prove certain results such as existence of solutions for certain problems like optimization problems in economics, engineering and other fields. It is also a fundamental theorem that is covered in most calculus textbooks.
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