Exercise · ZW-CONCAVITY-AND-INFLECTION-POINTS-003 · ★★★ Advanced
At which values of x does \(f(x)=x^{4} - 6 x^{2}\) have inflection points?
Choose one option.
Show answer
- A. \(x=-1,\ x=1\)
Explanation
Compute \(f''(x)=12 \left(x^{2} - 1\right)=12 \left(x - 1\right) \left(x + 1\right)\), which vanishes at \(x=-1,\ x=1\) and changes sign at both. They are the inflection points.
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-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-006 What is an inflection point of a graph?
- 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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