4. Problem 4 In the eld of ecology, one often wants to dene a measurement of ecological diversity. One common way of doing this is called the Shannon index. In the case where our enviromnent has three distinct species, the Shamron index is given by the function H($,'y,2) = mln (m) _ glut\") _ 2111(2) where each value of my, 2 (as a decimal between zero and one) represent the proportion of species 1, 2, and 3 - respectively. Generally speaking, a higher value of H represents a more ecological diverse community.1 Suppose we wanted to nd what values of 3:, y, and z maximize our ecological diversity. There are several different ways of doing this: we will explore two. (a) (1 Point) (Evaluated for mathematical work - accuracy and notational neatness) One way to approach this problem is to use the fact that m+y+ z = 1 as a constraint to reduce our function. Solving for 2 gives 2 = 1 m 1;, which reduces our function H to may) = -mh1(m)-y1n{y)(1my)m(1 zy). Find the absolute maximum of this function by lling in the details in the following argument. You may use the following facts without proof: 0 114953;) =ln[1xy)ln(m) yim,y)=h1(1-m-yJ-1n(y) _ y-l . zrlmry) _ 3(1 _ 2': _ y) 1 0 sz(m:y) = m ml ' HM\") = y(1 xy) Argument: The function H(2:, 3;) has one critical point: at (spy) = l:| . Algebra work: This critical point is a local maximum of the function, which can be justied with the Second Partials Test: Since there are no other critical points to speak of, this local maximum must also be an absolute maximum. Hence, the absolute maximum of the function H(m,y) is at the point [x,y) = (b) (1 Point) (Evaluated for mathematical work - accuracy and notational neatnms) However, we could also approach this problem with Lagrange Multipliers. This would require setting up a constrained optimization problem, with our objective function as H(m, y,z) = mh1{x) yln (y) 2111(2) and constraint x + y + z = 1. Set up {do not solve) the Lagrange equations (a system of four equations and four unknowns) to solve this problem. System of equations goes here: (c) (1 Point) (Evaluated for justification and written explanation) Which of these methods do you prefer? Which do you think is cleaner and easier to perform? Reflect on the pros and cons of each method. (I helped a little bit here with the differentiation and not asking for all of the algebra, but I think enough work has been done for you to have an opinion. This is a purely subjective question, and either position is justifiable) Explanation: (Use complete sentences)