First Derivative Test for Local Extrema
For the function $f(x)=x^3-3x^2+2$, the derivative is $f'(x)=3x^2-6x$, which yields critical points at $x=0$ and $x=2$. A researcher mistakenly concludes that a local maximum occurs at $x=2$. What is the error in this analysis?
A
The error is due to an incorrect factorization of the function f(x).
B
The mistake lies in the differentiation process; the derivative of f(x) should have an additional constant term.
C
There is no mistake; x = 2 is correctly identified as a local maximum.
D
The error is in the interpretation of the first derivative test; a proper sign analysis shows that x = 0 is the local maximum and x = 2 is a local minimum.
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