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2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem 2.3.1, (ii) by using Theorem 2.3.4.

2. (a) Let f be defined for all x, y by f(x, y) x-y - x. Show that f is concave (i) by using Theorem 2.3.1, (ii) by using Theorem 2.3.4.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you help with question 2? I attached a picture of the theorem to use. 

 

.. CHAPTER 2I MULTIVARIABLE CALCULUS 2. (a) Let f be defined for all x, y by f(x, y) x-y - x². Show that f is concave (i) by using Theorem 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 > 0 and 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 + 4ay SM 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)² SM 6. Suppose y = f(x) is a production function determining output y as a function of the vector x of 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. Is 902 BANG prt sc delete home pua Bd wnu lock -> backspace 8 [ home enter 4. ->
Transcribed Image Text:.. CHAPTER 2I MULTIVARIABLE CALCULUS 2. (a) Let f be defined for all x, y by f(x, y) x-y - x². Show that f is concave (i) by using Theorem 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 > 0 and 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 + 4ay SM 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)² SM 6. Suppose y = f(x) is a production function determining output y as a function of the vector x of 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. Is 902 BANG prt sc delete home pua Bd wnu lock -> backspace 8 [ home enter 4. ->
ydsaeter,Peter Hammond,Atle Seierstad,Ar.. 35 / 310 185% + 99 CHAPTER 2 I MULTIVARIABLE CALCULUS Concavity/Convexity for C2 Functions Checking the sign of the second derivative is often a quick way to decide where a C2 function of one variable is concave or convex. For functions of two variables there is a corresponding test which is often used (it is a special case of Theorems 2.3.2 and 2.3.3 below). THEOREM 2-3,1 Let 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 convex A 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 as well. (In fact, the second inequality implies fi f2 > (f2)² > 0. Thus if fi > 0, then f2 > 0 as well.) In a similar way, the two inequalities in part (d) imply that f < 0. MARUGK A R
Transcribed Image Text:ydsaeter,Peter Hammond,Atle Seierstad,Ar.. 35 / 310 185% + 99 CHAPTER 2 I MULTIVARIABLE CALCULUS Concavity/Convexity for C2 Functions Checking the sign of the second derivative is often a quick way to decide where a C2 function of one variable is concave or convex. For functions of two variables there is a corresponding test which is often used (it is a special case of Theorems 2.3.2 and 2.3.3 below). THEOREM 2-3,1 Let 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 convex A 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 as well. (In fact, the second inequality implies fi f2 > (f2)² > 0. Thus if fi > 0, then f2 > 0 as well.) In a similar way, the two inequalities in part (d) imply that f < 0. MARUGK A R
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