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The cubic function r(x) has a point of inflection at (-1,2). Suppose that r'(-1) = -2 and that...

Question:

The cubic function {eq}r(x) {/eq} has a point of inflection at {eq}(-1,2) {/eq}. Suppose that {eq}r'(-1) = -2 {/eq} and that {eq}r''(0) = -6 {/eq} and {eq}r''(-2) = 6 {/eq}.

Use the given information to decide whether the inflection point {eq}(-1,2) {/eq} is a:

(i) point of fastest increase;

(ii) point of slowest increase;

(ii) point of fastest decrease; or

(iv) point of slowest decrease.

Point of Inflection:

The point of inflection is where the graph changes concavity. The concavity of the function describes the shape whether it is curling up or curling down.

Answer and Explanation:

The point of inflection is at (-1,2). The slope is negative at the point. This means that the function is decreasing at this point. If at -2 the concavity is positive and at 0 the concavity is negative, the function went concave up to concave down. This means that the point of inflection is the point of slowest decrease. The answer is (ii).


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Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5
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