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


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).

Learn more about this topic:

Concavity and Inflection Points on Graphs

from Math 104: Calculus

Chapter 9 / Lesson 5

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