Exercise · ZW-CONCAVITY-AND-INFLECTION-POINTS-006 · ★☆☆ Basic
What is an inflection point of a graph?
Choose one option.
Show answer
- B. Where the concavity changes direction
Explanation
An inflection point is where the graph switches from concave up to concave down, or the reverse. The other options describe maxima, intercepts or zero slopes.
Review the concept: Concavity and inflection points
If \(f^{\prime\prime}>0\) on an interval, the graph is concave up; if \(f^{\prime\prime}<0\), concave down. An inflection point is where the concavity changes, and there \(f^{\prime\prime}(c)=0\) or does not exist.
\[\begin{gathered}f''>0\ \Rightarrow\ \text{concave up}\\ f''<0\ \Rightarrow\ \text{concave down}\end{gathered}\]
More practice problems: concavity and inflection points
- ZW-CONCAVITY-AND-INFLECTION-POINTS-001 At which value of x does f(x)=x^3 - 6 x^2 + 9 x have its inflection point?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-002 At which value of x does f(x)=x^3 - 3 x^2 + 1 have its inflection point?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-003 At which values of x does f(x)=x^4 - 6 x^2 have inflection points?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-004 For f(x)=x^4, f''(0)=0. Does f have an inflection point at x = 0?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-005 What does f''(x)>0 on an interval indicate?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-007 On which interval is the graph of f(x)=x^3 concave up?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-008 Where is the inflection point of f(x)=x^3-3x?
- ZW-CONCAVITY-AND-INFLECTION-POINTS-009 At which values of x does f(x)=x^4-4x^3 have inflection points?
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