APFIVE Experimental Procedure: In an optimization experiment, a student is asked to find the minimum of the function $$f(x)=x^3-6*x^2+9*x+15$$. The student first finds the critical points by setting $$f'(x)=3*x^2-12*x+9$$ equal to zero. However, when calculating the second derivative, the student incorrectly computes $$f''(x)=6*x-12$$ to equal 4 at a critical point (when it should be -2), and consequently concludes that the critical point is a minimum. What is the mistake in this procedure?
The error lies in using the second derivative test; instead, the student should rely on the first derivative test for functions of this form.
The mistake is that the function f(x) is cubic and does not have a true minimum, so any attempt to find one is inherently flawed.
The student should have set the first derivative equal to 1 instead of 0 to correctly identify the critical points for optimization.
The student made an arithmetic error in computing the second derivative at the critical point, leading to an incorrect conclusion about the concavity and hence the nature of the critical point.
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