Chapter 12: Q. 37 (page 989)

Extrema: Find the local maxima, local minima, and saddle points of the given functions.
$f (x, y) = 2 x^{2} + y^{2} + y + 5 .$

Short Answer

Expert verified

$Critical point as (0, - \frac{1}{2}) and the local minima point for the function is (0, - \frac{1}{2})$

Step by step solution

. Given

$f (x, y) = 2 x^{2} + y^{2} + y + 5 .$

Step 2. Finding saddle points.

$Given function is f (x, y) = 2 x ² + y ² + y + 5, which is a polynomial and hence differentiable . The gradient of this function is \nabla f (x, y) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j = 4 xi + (2 y + 1) j At the critical points the gradient of a function vanishes, that is \nabla f (x, y) = 0 . from the above the critical points are given by 4 x = 0 and 2 y + 1 = 0 . Solving these two gives one critical point as (0, - \frac{1}{2}) .$

Step 3. Finding maxima and minima .

$Now the second order derivatives of the given function are \frac{\partial^{2} f}{\partial x^{2}} = 4, \frac{\partial^{2} f}{\partial y^{2}} = 2, \frac{\partial^{2} f}{\partial y^{} \partial x} = 0 . Hence the discriminate of the given function is H_{f} (x, y) = \frac{\partial^{2} f}{\partial x^{2}} \frac{\partial^{2} f}{\partial y^{2}} - {(\frac{\partial^{2} f}{\partial y^{} \partial x})}^{2} = 4 (2) - 0 = 8 . Since, H_{f} (x, y) = 8 > 0 and \frac{\partial^{2} f}{\partial x^{2}} = 4 > 0, \frac{\partial^{2} f}{\partial y^{2}} = 2 > 0 Therefore the local minima point for the function is (0, - \frac{1}{2})$

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Extrema: Find the local maxima, local minima, and saddle points of the given functions.
$f (x, y) = 2 x^{2} + y^{2} + y + 5 .$

Short Answer

Step by step solution

. Given

Step 2. Finding saddle points.

Step 3. Finding maxima and minima .

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Extrema: Find the local maxima, local minima, and saddle points of the given functions.f(x,y)=2x2+y2+y+5.

Short Answer

Step by step solution

. Given

Step 2. Finding saddle points.

Step 3. Finding maxima and minima .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Pure Maths

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Study anywhere. Anytime. Across all devices.

Extrema: Find the local maxima, local minima, and saddle points of the given functions.
$f (x, y) = 2 x^{2} + y^{2} + y + 5 .$