APFIVE First Derivative Test For Relative Extrema
Which of the following correctly identifies the relative extrema of the function $$p(x)=x^3-6*x^2+9*x+2$$ based on the first derivative test?
A
The function possesses no relative extrema since the first derivative test applies only to functions with quadratic behavior.
B
Since $$p'(x)=3*(x-1)*(x-3)$$, the function has a relative maximum at $$x=1$$ (where the derivative changes from positive to negative) and a relative minimum at $$x=3$$ (where it changes from negative to positive).
C
The function has a relative maximum at $$x=3$$ and a relative minimum at $$x=1$$ because the derivative is zero at both points.
D
Relative extrema cannot be determined using the first derivative test because the derivative does not change sign at the critical points.
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