That the function f ( x , y ) = 1 3 x 3 + 2 3 y 3 / 2 − x y satisfies f ( x , y ) ≥ 0 ( x ≥ 0 , y ≥ 0 ) and find the critical points of the function. (a) We need to verify that the set of critical points of f is a parabola y = x 2 and that the second derivative test fails for these points. (b) We need to show that for fixed b , the function g ( x ) = f ( x , b ) is concave up for x > 0 with a critical point at x = b 1 / 2 . (c) We need to conclude that f ( a , b ) ≥ f ( b 1 / 2 , b ) = 0 for all a , b ≥ 0

Question

Want to see the full answer?

Answer 1

Question

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Solve : X + sin x = 0. By the false positioning numerical method

Solve: X + sin X = 0 by the false positionining numerical method

On from the equation: 2 u = C₁ + C₂ Y + Czy + Cu y³ Find C₁, C₂, C3 and Cy Using these following Cases : (a) 4=0 at y=0 (b) U = U∞ at y = 8 du (c) at Y = S ду --y. ди = 0 at y = 0 буг

Answer 2

Question

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Answer 6

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Answer 7

Chapter 15.7, Problem 25E

To determine

That the function $f (x, y) = \frac{1}{3} x^{3} + \frac{2}{3} y^{3 / 2} - x y$ satisfies $f (x, y) \geq 0 (x \geq 0, y \geq 0)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y = x^{2}$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$ , the function $g (x) = f (x, b)$ is concave up for $x > 0$ with a critical point at $x = b^{1 / 2}$ .

(c) We need to conclude that $f (a, b) \geq f (b^{1 / 2}, b) = 0$ for all $a, b \geq 0$

Answer 8

Expert Solution & Answer

Answer 9

Expert Solution & Answer

Answer 10

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 11

Ch. 15.1 - Prob. 1PQ Ch. 15.1 - Prob. 2PQ Ch. 15.1 - Prob. 3PQ Ch. 15.1 - Prob. 4PQ Ch. 15.1 - Prob. 5PQ Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - Prob. 3E Ch. 15.1 - Prob. 4E Ch. 15.1 - Prob. 5E

Solution Summary: The author determines that the function f (x, y) = 1 3 x 3 + 2 3

Concept explainers

Videos

Want to see the full answer?

Chapter 15 Solutions