Wize AP Calculus (BC) Textbook > Applications of Differentiation: Analytical

The Extreme Value Theorem

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Extreme Value Theorem (EVT)
Continuous functions on a closed interval are guaranteed to have a maximum and a minimum. 
Extreme Value Theorem
If f(x)f\left(x\right)f(x) is a continuous function on [a,b]\left[a,b\right][a,b], then f(x)f\left(x\right)f(x) must have both a maximum and minimum value on [a,b]\left[a,b\right][a,b].



Formally, there existsm,M∈[a,b]m,M\in[a,b]m,M∈[a,b] such that f(M)≥f(x)≥f(m), for all x∈[a,b]f(M)\ge f(x)\ge f(m)
\text{, for all}\ x\in[a,b]f(M)≥f(x)≥f(m), for all x∈[a,b]