IVT

Topic: Intermediate Value Theorem Rule Of Four Math By: Notes

Verbal

Intermediate Value Theorem:

If a function, f(x),

is continuous on the closed interval [a, b] and

y is a value between f(a) and f(b),

Then

There is a number c in the interval (a, b) for which f(c) = y

Corollary: Extreme Value Theorem:

If a function, f(x),

is continuous on the closed interval [a, b]

Then

There are numbers c₁ & c₂ in the interval (a, b) for which f(c₁) & f(c₂) are the max & min for f(x) on that interval.

Analytical

Intermediate Value Theorem:

If a function, f(x),

is continuous for a < x < b and

Then

a < c < b

f(a) < f(c) < f(b)

Corollary: Extreme Value Theorem:

If a function, f(x),

is continuous for a < x < b and

Then

a < c₁ < b & a< c₂ < b

f(c₁) = MAX(f(x)) & f(c₂) = MIM(f(x))

Numerical

Graphical

If ( a < c < b ) then ( f(a) < f(c) < f(b) )

( f(a) > f(c) > f(b) )

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