Unit 5 Theorems and L'Hopital's Rule

EVT, MVT, Rolle's

Relative extrema

Point where function changes from + → - or - → +

Absolute extrema

Point on a graph with greatest or least y-values

Extreme Value Theorem

If f(x) is continuous on [a, b], then there exists an absolute maximum and an absolute minimum on [a, b] at either x = a, x = b, or any value of x on (a, b) such that f'(x) == 0 or f'(x) is undefined (Critical Numbers)

Where to find Absolute extrema by EVT

• f(a)

• f(b)

• Critical Numbers (Find 1st Derivative and set it to 0)

Rolle’s Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b) and f(a) = f(b), then there must exist at least one value, x = c, on (a, b) such that f'(c) = 0

What conditions must be true for Rolle’s Theorem to be applicable?

• f(x) is continuous on [a, b]

• f(x) is differentiable on (a, b)

• f(a) = f(b)

Mean Value Theorem

If f(x) is continuous on [a, b] and differentiable on (a, b), then there must exist a value, x= c on (a, b) such that f'(c) = f(a) - f(b) / a - b

What conditions must be true for Mean Value Theorem to be applicable?

• f(x) must be continuous on [a, b]

• f(x) must be differentiable on (a, b)

L’Hopital’s Rule

If lim f(x)/g(x) = 0/0 or lim f(x)/g(x) = +- infinity / +- infinity

x → a x → a

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then lim f(x)/g(x) = lim f’(x)/g'(x)

x → a x → a