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Today's calculation of Integral 685

Source:

February 26, 2011
calculusintegrationfunctioncalculus computations

Problem Statement

Suppose that a cubic function with respect to xx, f(x)=ax3+bx2+cx+df(x)=ax^3+bx^2+cx+d satisfies all of 3 conditions: f(1)=1, f(−1)=−1, ∫−11(bx2+cx+d) dx=1f(1)=1,\ f(-1)=-1,\ \int_{-1}^1 (bx^2+cx+d)\ dx=1.
Find f(x)f(x) for which I=∫−112{f′′(x)}2 dxI=\int_{-1}^{\frac 12} \{f''(x)\}^2\ dx is minimized, the find the minimum value.
2011 Tokyo University entrance exam/Humanities, Problem 1
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