How to find Critical Points & Classify, Exercises of Calculus

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Find he ctl point of the ntion Ten ashe avn pert ve Test determine whether it = minimum, savin, or ee pink Key) 60? + 2y? — ay tx Pat ofa Recall the cetinition ct a ertical point. Apaint = (2, 5) im the domain of f(x, /) is a critical point if f(2, ) fo v or §,(a, 6) doce not exist and if fy{2, 6) = Bg) or Fy22. 8) does net exist. Find the firstorder partial derivatives #60, y) and toon = [Dey 4] yl ionelw-s | aaa] ¥) Pan? of 4 ‘Set the pertal derivanives equal to zero and solve the system of equations for tre ertical points, since the partial cerivatives arc polynomials, they exist everywhere. Hence the solutions to se system of cquatione will yicls al the pozsible critical soints. ve yitea ay Keg Tre cailicel point is tx, ¥) Pan 3of4 Sine is the only solution to the system af esjiations, itis the only evitieal point. Recall the Sccund Derivative Test for ciesefying the type af erttcal point. Lal P= (9, b) be a uiilical point of fy, 7), Assun AP D(a, 8 > 0 std Lf, b) > 0, Une Ray BY foo fy fey ats conlinuoas rigar P sind Uhel U's deri FE seat inna]. SIF DG. 6)> 0 and f(a, B) < V, then Aa, b) is [a local maximum » | [a local maximum]. +1 9(9,8) © 0, sn Ka, baa aT (9,8) —0, Lown Une sts RES ]™ rennet] Find Une sucuind-order partial vst NBG, ¥) Be Oly V9 — Fes Wyle VY ~ Fai v. The 1 Goo Sy lop atid use Uren Lo comaute OCs, ye Fede fhe befeev) veyy= dé A Para otd using 20+, £7 and fats ¥) ~ 42 found inpart 3, evaluate of 2, ~ 7 ae) =| Ax arn ay) = [2 AY inw Of--+,-L) aa >] ee o(-.--4) Sev El * of £,{-+,- 4) \~ > 0, ie poin oS 2) BEY [EF 2 ome