Curve sketching practice question 30
A student finds the inflection points of the function below as shown. Find the first line that is wrong.
is where an inflection point can be, not where it must be. What else has to happen there?
Check the sign of just to the left and just to the right of each zero.
Theory UBC Math 100 §7.3CLP-1 §3.6Key results
Corrected working
Strategy
Find where f″ is zero, then check that f″ changes sign there.
Working
Second derivative, factored.
x² is never negative, so only the factor x − 3 changes the sign.
The concavity changes at 3 and not at 0.
Answer
The answer is the only inflection point is at x = 3.
Check
Near x = 0, f behaves like −5x4, an upside-down bowl on both sides, so its concavity cannot change there.
Where this goes wrong
Calling every zero of f″ an inflection point. At x = 0 the second derivative touches zero without changing sign, so the concavity does not change.
The idea behind it
The curve sketching lesson, a figure to move, in a few steps
Leans on
The second derivative, from Differentiation rules
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