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5.6 Determining Concavity of Functions over Their Domains

Syllabus
2020
Topic
5.6
Level
—

Learning objectives

—FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.6• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives.• FUN-4.A.4 The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.• FUN-4.A.5 The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.• FUN-4.A.6 The second derivative of a function may be used to locate points of inflection for the graph of the original function.• Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.

Read Concavity from Changing Slopes

Concavity describes how the slopes of ff change. If f′f' is increasing, successive tangent slopes become larger and ff is concave up. If f′f' is decreasing, the slopes become smaller and ff is concave down. Because f′′f'' measures the rate of change of f′f', its sign gives the same information directly.

Behavior of f′f' Sign of f′′f'' Concavity of ff
increasing f′′(x)>0f''(x)>0 concave up
decreasing f′′(x)<0f''(x)<0 concave down

Find domain values where f′′(x)=0f''(x)=0 or f′′(x)f''(x) does not exist. Use them, together with any domain breaks, to divide the domain into open intervals. Determine the sign of f′′f'' on each interval. A point on the graph is an inflection point only when the concavity changes across it.

For f(x)=x3−3x2f(x)=x^3-3x^2, f′′(x)=6x−6=6(x−1).f''(x)=6x-6=6(x-1). This is negative for x<1x<1 and positive for x>1x>1, so ff is concave down on (−∞,1)(-\infty,1) and concave up on (1,∞)(1,\infty). The concavity changes at x=1x=1, and f(1)=−2f(1)=-2, so (1,−2)(1,-2) is an inflection point.

The equation f′′(c)=0f''(c)=0 or an undefined second derivative identifies only a candidate. Without a change from concave up to concave down or vice versa, there is no inflection point.