Splitting cases

Calculus Level 3

\[ \large f(x) = \sin^3(x) + k \sin^2(x) \]

For variable \(k\) independent of \(x\), consider the function \(f\) on the interval \( -\frac\pi2 < x < \frac\pi2 \) as described above. Find the sum of all integral values of \(k\) such that \(f(x) \) has exactly one minimum value and one maximum value.

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