Second Derivative Test Error Analysis
A student attempts to find a minimum of the function . After finding the critical points by setting to zero, the student applies the second derivative test. At one critical point, the student incorrectly calculates the value of to be 4, when the correct value is -2. Based on this error, the student concludes that the critical point corresponds to a minimum. What is the mistake in the student’s reasoning?
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 student should have set the first derivative equal to 1 instead of 0 to correctly identify the critical points for optimization.
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 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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