24 v=H'().R a(v, w) = +vw] dx f(x) = prove that ta(vi) vk, wil If f (22/ Let V = Ho' (v) = Ho' (I) I= (0,1) a(u,v) = Sul.vidu Fevs =) B.vdx. prove that I acu,v)] & hull, "I wil Q3/ Let V=Ho (u). USIR a(v, w) = Vv w x L(v) = (s-v dx f- prove that a(vw) is v- elliptic Qu/ consider the following boundary value problem Juu =f, x6 I = (0,1) Find variational form. ②F. E. solution. prove that the bilinear form al.,-) Continouse, v- elliptic.

2b). Let y' + 5x?y²+ yxt = 624. A) Find y'

Show that (∂z/∂x)2 + (∂z/∂y)2 = (∂z/∂r)2 + (1/r2) (∂z/∂θ)2

4. Let f (x) = x³ -x² + 5. a) Find the y-intercept of f. y-intercept: b) Find f' and f", and determine where each are 0 and/or do not exist (DNE). If none, write "none". f' = 0: f' DNE: f" = 0: f" DNE: c) E Do a sign analysis on f' and f". d) Find the intervals on which f is increasing and decreasing. Increasing: Decreasing: e) Find the intervals on which f is concave up and concave down. Concave up: Concave down: f) answers as (x, y) points. Find all local maxima, local minima, and inflection points of f. Be sure to write your Local max: Local min: Inflection point(s): -4 -3 -1 g) Sketch the graph of f.