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Justifying Maxima and Minima Justifying Maxima and Minima By John Losse - updated 12/18/13 Free response questions frequently ask the student to justify why a relative maximum, minimum or inflection point occurs. The grading standard for the "justify point" is very strict but easy to attain if the student is careful and learns the required language. Important! While a sign chart might help the student decide whether there is a max, min, etc. the sign chart is not a justification. The student must use words (preferably in a grammatically correct sentence). Further, the student must clearly remove any ambiguity about whether the statement is referring to the function itself, its derivative or its second derivative, because typically more than one of these is in play in the problem. Additionally, the student should stick with mathematically precise terms such as "f'(x) >0", "is increasing", "is zero", "is positive" etc. instead of less precise phrases like "going up", "gets bigger", "transitions from", "reverses". In all cases, "it" or "its" must have a clear antecedent. In every case it is better to replace "it" by whatever "it" stands for. Example: Bad: It's increasing because it is positive Good: f is increasing because f'(x) is positive Here is a problem showing the basic idea. The figure shows the graph of a function f(t) defined for -10 ≤ t ≤ 10. The function g is x defined by g(x) = f (t)dt . ò-5 IMPORTANT: Before the student does anything else in this problem the student must write that "g' = f ", "g'(x) = f(x)", or the clear equivalent. If g" will be used, g" = f ' or the equivalent should also be written – immediately! Here are "bad" (i.e. insufficient) justifications for "g has a relative minimum" at x= -2. Comments are in italics. INSUFFICIENT JUSTIFICATIONS: g goes from decreasing to increasing at x=-2 (of course – that's what always happens at a min!) The slope goes from negative to positive at x=-2 (slope of what?) The derivative goes from negative to positive at x=-2 (derivative of what?) f crosses the x-axis at x=-2 (the student needs to say what's so special about that!) It changes sign from negative to positive at x=-2 (All the student needs to do is replace "It" with " g ' ".) The derivative is increasing at x=-2 (Derivative of what? Why does this matter?) The velocity of g goes from negative to positive at x=-2 no comment g reverses direction at x = -2 x SUFFICIENT JUSTIFICATIONS again, g(x) = f (t)dt . ò-5 Here are sufficient justifications for "g has a relative minimum at x = -2" (remember that the student must have previously stated that g' = f; if not, this must be part of the justification phrase.) g' goes from negative to positive at x = -2. Better: g' = f goes …. (and similarly throughout!) the slope of g of goes from negative to positive at -2 the derivative of g goes from negative (g decreasing) to positive (g increasing) at x=-2 g' changes sign from negative to positive at -2 g' goes from negative to positive at -2 g' is negative just before -2 and positive just after -2. g'(-2) = 0 and g' is increasing, so g"(-2)>0, so g has a minimum by the Second Derivative Test. Remember, a sign chart is not enough. Readers are instructed to ignore a sign chart completely when deciding if a justification is sufficient. Justification of the Inflection Point at x = 3 Important: Before making any reference to g", the student must have stated that g' = f and, (ideally) that g" = f ' . INSUFFICIENT JUSTIFICATIONS g"(3) = 0 (it doesn't, and even if it did . ) g"(3) does not exist g goes from concave up to concave down at x=3 (what about the graph shows this?) SUFFICIENT JUSTIFICATIONS g' = f goes from increasing (g concave up) to decreasing (g concave down) at 3 g" = f ' goes from positive to negative at 3 g" = f ' changes sign from positive (g is concave up) to negative (g is concave down) at x=3. g" is positive just before x=3 and negative just after 3. Other things your students need to know about max/min and IP justifications: Even a well-labeled and accurate graph of g is insufficient! For example, what features of the graph of f near x=-2 or x = 3 guarantee that the student's graph of g is accurate there? The student would need to explain this in words, and these words would themselves (without the graph) earn the point. The graph might help the student figure out where the minimum is, but will not be sufficient for the justification point. .
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