4. For what values of the constant a is the following function concave/convex? f(x, y)= -6x2 + (2a+4)xy-y+4ay

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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
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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. ->
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