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Rolle’s Theorem and the Mean Value Theorem
Guaranteeing Extrema
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Rolle’s Theorem Guarantees the existence of an extreme value in the interior of a closed interval
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Rolle’s Theorem Let f be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f’(c) = 0.
![Rolle’s Theorem Let f be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b). If.](http://slideplayer.com./slide/16481933/96/images/3/Rolle%E2%80%99s+Theorem+Let+f+be+a+continuous+function+on+the+closed+interval+%5Ba%2C+b%5D+and+differentiable+on+the+open+interval+%28a%2C+b%29.+If..jpg "f(a) = f(b) then there is at least one number c in (a, b) such that f’(c) = 0.")
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Rolle’s Theorem Three things that must be true for the theorem to hold: (a) the function must be continuous (b) the function must be differentiable (c) f(a) must equal f(b)
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Rolle’s Theorem Examples
Book On-line
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Mean Value Theorem If f is continuous on the closed interval
[a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that
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Mean Value Theorem In other words, the derivative equals the slope of the line.
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Mean Value Theorem Tutorial Examples More Examples
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