Q. 37 Extrema: Find the local maxima, ... [FREE SOLUTION]

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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.

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

Step 3. Finding maxima and minima .

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

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91影视

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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91影视

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

Discrete Mathematics

Applied Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

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