Can you help with question 2? I attached a picture of the theorem to use. ..CHAPTER 2I MULTIVARIABLE CALCULUS2. (a) Let f be defined for all x, y by f(x, y) x-y - x². Show that f is concave (i) by usingTheorem 2.3.1, (ii) by using Theorem 2.3.4.(b) Show that -e-Ju.y) is concave.3. (a) Show that f (x, y) = axr² + 2bxy + cy² + px +qy+r is strictly concave if ac – b² > 0and a < 0, whereas it is strictly convex if ac - b? > 0 and a > 0.(b) Find necessary and sufficient conditions for f(x, y) to be concave/convex.4. For what values of the constant a is the following function concave/convex?f(x, y) =-6x²+ (2a + 4)xy – y² + 4aySM 5. Examine the convexity/concavity of the following functions:á+x³ – ,? – 6 + x = 2 (x)(b) z = e*+y + e*=y - }y(c) w = (x+ 2y + 3z)²SM 6. Suppose y = f(x) is a production function determining output y as a function of the vector xof nonnegative factor inputs, with f(0) = 0. Show that:(a) If f is concave, then f(x) <0 (so each marginal product f{(x) is decreasing).(b) If f is concave, then f (Ax)/A is decreasing as a function of 2.(c) If f is homogeneous of degree 1 (constant returns to scale), then f is not strictly concave.Tat f he defined for all v in R" by f(x) = |x H =./x +...+x². Prove that f is convex. Is902BANGprt scdeletehomepuaBdwnulock->backspace8[homeenter4.-> ydsaeter,Peter Hammond,Atle Seierstad,Ar..35 / 310185%+99CHAPTER 2 I MULTIVARIABLE CALCULUSConcavity/Convexity for C2 FunctionsChecking the sign of the second derivative is often a quick way to decide where a C2 functionof one variable is concave or convex. For functions of two variables there is a correspondingtest which is often used (it is a special case of Theorems 2.3.2 and 2.3.3 below).THEOREM 2-3,1Let z = f (x, y) be a C² function defined on an open convex set S in the plane.Then (all inequalities must hold throughout S):(a) ƒ is convexA f 2 0, fm 2 0, and fij fn - (f)? > 0.(b) f is concave - fi s0, f s0, and fi f- (S)? > 0.(c) f > 0 and f, fm- (f)? > 0 f is strictly convex.(d) f < 0 and fi fm- (f)² > 0 = f is strictly concave.NOTE 2 The implications in parts (c) and (d) cannot be reversed. For example, f (x, y) =x*+y* is strictly convex in the whole plane, even though f (0, 0) = 0. (See Problem 2.4.6.)NOTE 3 From the two inequalities specified in part (c), it follows that f (x, y) > 0 aswell. (In fact, the second inequality implies fi f2 > (f2)² > 0. Thus if fi > 0, thenf2 > 0 as well.) In a similar way, the two inequalities in part (d) imply that f < 0.MARUGK A R

Name: Can you help with question 2? I attached a picture of the theorem to u
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Description: Can you help with question 2? I attached a picture of the theorem to use. ..CHAPTER 2I MULTIVARIABLE CALCULUS2. (a) Let f be defined for all x, y by f(x, y) x-y - x². Show that f is concave (i) by usingTheorem 2.3.1, (ii) by using Theorem 2.3.4.(b) S