if given f(x), the x coordinates of where max/min are occuring will be (__) on the _____ graph. (points on it’s x axis)
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(x,0)
x-coordinates of points of inflection will be ____ on the **f”(x)** graph (points on the x-axis of f”)
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when it looks like a U and has positive values on the f”(x) graph
given f(x), when is the function concave up?
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when it looks like an upside down U and when f”(x) is negative
given f(x), when is the function concave down? (In terms of f”(x))
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the x-coordinate will touch the x axis but not go through on the graph of f’(x)
given f(x), when there is a point on the function where the slope is zero but is NOT a max/min, _____________________.
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when f(x) changes from concave up to concave down (or vice versa)
given f(x), when do points of inflection occur?
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max/min…. sign change…. max/min
given f’(x), check the x-intercepts because they may be where the _____ of f(x) occur. there must be a _______ ______ for it to be a _____
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positive
given f’(x), look for intervals where all of the y-values are ________. f(x) will be increasing on these intervals
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negative
given f’(x), look for intervals where all of the y-values are _______. f(x) will be decreasing on these intervals
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positive
given f’(x), intervals where f’(x) is increasing will show _________ y-values on the f”(x) graph
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negative
given f’(x), intervals where f’(x) is decreasing will show ________ y-values on the f”(x) graph
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concave up
given f’(x), intervals where f’(x) is incresasing means f(x) is _____________
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concave down
given' f’(x), intervals where f’(x) is decreasing means f(x) is ______________
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points of inflection…. points of inflection
if given f”(x), check x-intercepts. the x coordinates are the x coordinates of the _____________. if there are no x-intercepts, then there are no ______________.
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positive
if given f”(x), look for intervals where all the y-values are _________. f’(x) will be increasing on those intervals
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negative
if given f”(x), look for intervals where all the y-values are __________. f’(x) will be decreasing on those intervals.
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increasing
if f’(x) and f”(x) have all positive values, then f(x) is _________. (think exponential functions)
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decreasing
if f’(x) and f”(x) have all negative values, then f(x) is __________. (think negative exponential functions)
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closed intervals
when answering a question along the lines of “on what intervals is f increasing/decreasing?” what kinds of intervals are you using?
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open intervals
when answering a question along the lines of “on what intervals is f concave up/down?” what kind of interval are you using?