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1966Balanced Softplus Tradeoff 21The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e x). What x is optimal?数学简单数值题未尝试免费1967Positive Softplus Tilt 22The linear reward is strong relative to the saturation penalty, so the optimum should be positive. The desk maximizes K(x) = 3 x - 4 ln(1+e x). What x is optimal?数学简单derivation未尝试免费1968Negative Softplus Tilt 23The saturation penalty dominates until x turns sufficiently negative. The desk maximizes K(x) = 1 x - 3 ln(1+e x). What x is optimal?数学中等derivation未尝试免费1969Which Side of Zero Is Optimal in the Linear-Softplus Problem 24For K(x)=m x - n ln(1+e x) with 0<m<n/2, is the unique optimizer positive, zero, or negative?数学中等derivation未尝试免费1970Maximum Value at the Interior Utility-Impact Optimum 25For F(x)=4 ln(1+x)-x 2 on x>-1, what x maximizes F and what is the maximum value?数学困难derivation未尝试免费