Question

Question Optimizing single Argument Function The general form of the League of Legends example (pages 145, 148) and sim- ilar cases such as the Software Development study involves four constant values which we will denote using {k,r,a, B}. The function to be optimized being E(x) = axk - Boxk+r For example in the League of Legends analysis we had k = r = 1, a = (1800 + T)/1800 (T being the initial number of tokens) and B = 2.5/1800. 1. Assuming that k > 1, r > 0, and both a > 0,3 > 0, derive the expression describing E'(x). That is find axk - Bæk+r 2. Identify the critical point(s) of E'(x) (as functions of the constants {k,r,a, B}). 3. Determine whether these are minima or maxima by applying the second derivative test, having constructed E"(x). As detailed as possible please!

Answer 1

(Pago No Data Page No . K?), roxxo, ß> 0 Dato р Ein) = 2 4 - 6 Kart - pkt - El(n) = 4kn E) карка) ) ] To find critical points - We equale E (l) ao .: Elix)=0 the EaK - PIK+ x) = 0 =) nk to or a K-Blkar)n = 0 = A=0 or Blktrl=XK mak B1K+) 7 1=o or t = KANT are critical n=0 / qc Blk+r) points of En

3 for second derivative test, we K+r- E chl= akup (k+r) n K+ 7-2 E"0) - KK CK+) X=BCK+Y) (k+Y-DX Ktr-2 Elx)= ak(K-1) x - B(K+NK+Y-1) a case 1 (6) = 0 test fails in this case K -2 ak LK CK-UP B (kery B (kty) Kira - pli+Y)(k+x=136 KM) XK(K-1) [ LK - (pikt/ Ixktr-2 BCK+ K+Y-D K-2 B / K+DCK+Y-DIXK XK(K-1) LKY [B1K+r)! Apiker

Pago No Date Pago No Date 2K(K-D) - plk+1(KEY- D LK Blkar) K BCK+ K-2 lak Tr LK (R-1) - BCK+D (K+Y-1) ak B (K + 1). A Blk+r) K-Y 2-1) – (K-1 KK LACK+n & K - LK-LK - LKrtOK Dk+ tack+o) XK B(kar) & krlak Blkar) Now we are given that kal,,12,0 830 830 which gives us JB (Kar)

(Page No Date Page No Date By second cheivative test En) is maximum when BOK+r).