Section 4.2

Section 4.2 - maximum of f. If f '(x) < 0 for...

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Section 4.2: The Mean Value Theorem Rolle's Theorem. Let f be a function which is differentiable on the closed interval [a, b]. If f(a) = f(b) then there exists a point c in (a, b) such that f '(c) = 0. Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that Corollary. Let f be a differentiable function such that the derivative f ' is positive on the closed interval [a, b]. Then f is increasing on [a, b]. Let f be a differentiable function such that the derivative f ' is negative on the closed interval [a, b]. Then f is decreasing on [a, b]. First Derivative Test. Suppose that c is a critical point of the function f and suppose that there is an interval (a, b) containing c. If f '(x) > 0 for all x in (a, c) and f '(x) < 0 for all x in (c, b), then c is a local
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Unformatted text preview: maximum of f. If f '(x) &lt; 0 for all x in (a, c) and f '(x) &gt; 0 for all x in (c, b), then c is a local minimum of f. Example: Determine the intervals on which the function is increasing or decreasing and the local maximums and local minimums. Since the domain of f is the same as the domain of f', 4 is the only critical number of f. Testing: By the First Derivative Test, x = 4 is a local minimum. Example: Since the domain of f is the same as the domain of f', -3 and 6 are the only critical numbers of f. Testing: x &lt; -3 f'(-10) = 672 f is increasing -3 &lt; x &lt; 6 f(0) = -108 f is decreasing x &gt;6 f(10) = 312 f is increasing By the First Derivative Test, x = -3 is a local minimum and x = 6 is a local maximum. x &lt; 4 f'(0) = -8 f is decreasing x &gt; 4 f'(5) = 2 f is increasing...
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Section 4.2 - maximum of f. If f '(x) &amp;amp;lt; 0 for...

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