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The Maximum Value

Source: 1996 National High School Mathematics League, Exam One, Problem 5

March 3, 2020
function

Problem Statement

On [1,2][1,2] if two functions f(x)=x2+px+qf(x)=x^2+px+q and g(x)=x+1x2g(x)=x+\frac{1}{x^2} get their minumum value at the same point, then the maximum value of f(x)f(x) on [1,2][1,2] is (A)4+11223+43(B)4−5223+43\text{(A)}4+\frac{11}{2}\sqrt[3]{2}+\sqrt[3]{4}\qquad\text{(B)}4-\frac{5}{2}\sqrt[3]{2}+\sqrt[3]{4} (C)1−1223+43(D)\text{(C)}1-\frac{1}{2}\sqrt[3]{2}+\sqrt[3]{4}\qquad\text{(D)} none above
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