Finding the Extrema of a FunctionDate: 10/25/96 at 7:28:28 From: Anonymous Subject: Extrema Dear Dr. Math, I have to analyze the following function: f(x)=(ln(1+exp(-a-bx-cx^2)))/((x+d)^2) Questions: - If -b/2c>d: is it possible to obtain an analytical expression for the maximum (if it exists) ? - Is it possible to calculate when f(d)=f(maximum) ? I have already determined that without 1/((x+d)^2) I can obtain the maximum, x=-b/2c. I hope that I have been clear. Thanks and kind regards, Henk Huinink Date: 11/05/96 at 21:07:46 From: Doctor Lorenzo Subject: Re: Extrema First of all, there is no global maximum, since as x approaches -d the function blows up. So we'll look for local maxima. Let's think of your function as the product of two (non-negative) factors, A = ln (1 + exp (-a-bx-cx^2)) and B = (x+d)^{-2}. As you already worked out, the factor A is increasing for x < -b/2c and decreasing for x> -b/2c. B, meanwhile, is increasing for x<-d and decreasing for x> -d. The only way to have a local maximum is to have one factor increasing while the other is decreasing, so any maximum (if it exists) has to occur between -d and -b/2c. This is true regardless of whether -d > -b/2c or -d < -b/2c. We treat the cases one at a time. First assume -d > -b/2c. At x = -b/2c the function is increasing, as A is stationary and B is increasing. As x approaches -d (from below), the function is certainly increasing, and B is blowing up while A isn't going to zero. So there are two possibilities: If c is large enough, you will get one maximum and one mimimum. A will decrease rapidly fairly soon after you get away from x=-b/2c, but B will increase substantially only later. However, if c is small, then A B' will be larger (in magnitude) than A' B throughout the interval, and you will have no maxima and no minima. The case -d < -b/2c is similar. For x slightly greater than -d, the derivative is sharply negative. For x near -b/2c, the derivative is somewhat negative. If c is large enough, the derivative will change signs twice in the intermediate region, while if c is small the derivative will stay negative throughout the region. I don't see any real hope of finding a closed-form expression for the location of the extrema, but perhaps somebody more clever than I can come up with one. -Doctor Lorenzo, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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