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8 Cards in this Set
- Front
- Back
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A curve is convex if... |
f’’ (x) > 0, for all values of x. |
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A curve is concave if... |
f’’ (x) < 0 , for all values of x. |
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What is a ‘point of inflection’ ? |
A point where the curve changes between concave and convex (i.e. f’’(x) changes between positive and negative) At a point of inflection: f’’(x) = 0 |
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However be careful, Not all points where f’’(x) = 0 are points of inflection. f’’(x) has to change from positive to negative or negative to positive either side of the point - to be a point of inflection. |
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What kind of stationary point would you have if... f’(x) = 0 and f’’(x) > 0 |
A minimum |
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What kind of stationary point would you have if... f’(x) = 0 and f’’(x) < 0 |
Maximum |
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What kind of stationary point would you have if... f’(x) = 0 and f’’(x) = 0 |
Any of the 3: max, min, or stationary point of inflection You have to look at f’’(x) either side: • if f”(x) > 0 either side of the point -> minimum • if f”(x) < 0 either side of the point -> maximum • if f”(x) changes sign either side of a point -> stationary point of inflection |
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Note that there is a difference between stationary points of inflections and points of inflections |
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