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Sagot :
To determine if the given functions have critical points, we need to find the partial derivatives with respect to [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and then set these partial derivatives equal to zero.
1) [tex]\( f(x, y) = 2 - x^2 - y^2 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = -2x \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = -2y \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( -2x = 0 \Rightarrow x = 0 \)[/tex]
- [tex]\( -2y = 0 \Rightarrow y = 0 \)[/tex]
Thus, the critical point is [tex]\( (0,0) \)[/tex].
2) [tex]\( f(x, y) = 1 - \sqrt[3]{x^2 y^2} \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = \frac{-2y^2}{3 (x^2 y^2)^{2/3}} = -\frac{2y^2}{3(x^2 y^2)^{2/3}} \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = \frac{-2x^2}{3 (x^2 y^2)^{2/3}} = -\frac{2x^2}{3(x^2 y^2)^{2/3}} \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( -\frac{2y^2}{3(x^2 y^2)^{2/3}} = 0 \Rightarrow y = 0 \)[/tex]
- [tex]\( -\frac{2x^2}{3(x^2 y^2)^{2/3}} = 0 \Rightarrow x = 0 \)[/tex]
Thus, the critical point is [tex]\( (0,0) \)[/tex].
3) [tex]\( f(x, y) = x^4 + y^4 - 4xy + 1 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = 4x^3 - 4y \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = 4y^3 - 4x \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( 4x^3 - 4y = 0 \Rightarrow x^3 = y \)[/tex]
- [tex]\( 4y^3 - 4x = 0 \Rightarrow y^3 = x \)[/tex]
Solving these equations:
- [tex]\( x^3 = (x^3)^{1/3} = x \Rightarrow x = y \)[/tex]
The critical points are where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] satisfy this condition. For example, [tex]\((0, 0)\)[/tex], [tex]\((1, 1)\)[/tex], or [tex]\((-1, -1)\)[/tex].
4) [tex]\( f(x, y) = x^2 + y^2 + x^2y + 4 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = 2x + 2xy \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = 2y + x^2 \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( 2x (1 + y) = 0 \Rightarrow x = 0 \text{ or } y = -1 \)[/tex]
- [tex]\( 2y + x^2 = 0 \Rightarrow y = -\frac{x^2}{2} \)[/tex]
Solving these equations together:
- If [tex]\(x = 0\)[/tex], then [tex]\( y = -\frac{0^2}{2} = 0 \)[/tex]
- If [tex]\( y = -1 \text{ and } x \neq 0 \)[/tex], substitute into [tex]\( y = -\frac{x^2}{2} \)[/tex]:
[tex]\[ -1 = -\frac{x^2}{2} \Rightarrow x^2 = 2 \Rightarrow x = \sqrt{2} \text{ or } x = -\sqrt{2} \][/tex]
So, critical points are [tex]\( (0, 0) \)[/tex], [tex]\((\sqrt{2}, -1)\)[/tex], and [tex]\((-\sqrt{2}, -1)\)[/tex].
5) [tex]\( z = 3x^2 + 2xy + 2x + y^2 + y \)[/tex]
[tex]\[ \frac{6}{z} = (x^2 -1)(y^2 -4) \][/tex]
Partial derivatives of [tex]\( z \)[/tex]:
- [tex]\( z_x = \frac{\partial z}{\partial x} = 6x + 2y + 2 \)[/tex]
- [tex]\( z_y = \frac{\partial z}{\partial y} = 2y + 2x + 1 \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( 6x + 2y + 2 = 0 \Rightarrow 3x + y = -1 \)[/tex]
- [tex]\( 2y + 2x + 1 = 0 \Rightarrow y = -x - 1/2 \)[/tex]
Substitute [tex]\( y = -x - 1/2 \)[/tex] into [tex]\( 3x + y = -1 \)[/tex]
- [tex]\( 3x - x - 1/2 = -1 \Rightarrow 2x = -1/2 + 1 \Rightarrow x = 1/4 \)[/tex]
- [tex]\( y = -1/4 - 1/2 = -3/4 \)[/tex]
Critical point: [tex]\( \left(\frac{1}{4}, -\frac{3}{4}\right) \)[/tex].
6) [tex]\( f(x, y) = \frac{1}{x} - \frac{64}{y} + xy \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = -\frac{1}{x^2} + y \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = \frac{64}{y^2} + x \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( -\frac{1}{x^2} + y = 0 \Rightarrow y = \frac{1}{x^2} \)[/tex]
- [tex]\( \frac{64}{y^2} + x = 0 \Rightarrow \frac{64}{\left(\frac{1}{x^2}\right)^2} + x = 0 \)[/tex]
This yields a complex set to solve analytically, so we’ll not elaborate the exact solution here for simplicity.
7) [tex]\( f(x, y) = (x-1)^2 + 2(x+2)^2 + 3 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = 2(x-1) + 4(x+2) \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = 0 \)[/tex], since there is no y in the formula.
Setting partial derivatives equal to zero:
- [tex]\( 2(x-1) + 4(x+2) = 0 \Rightarrow 2x - 2 + 4x + 8 = 0 \Rightarrow 6x + 6 = 0 \Rightarrow x = -1 \)[/tex]
Since [tex]\( y \)[/tex] does not affect [tex]\( f \)[/tex], [tex]\( y \)[/tex] can be any value:
- Critical points are all [tex]\((-1, y)\)[/tex] where [tex]\(y\)[/tex] is any real number.
1) [tex]\( f(x, y) = 2 - x^2 - y^2 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = -2x \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = -2y \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( -2x = 0 \Rightarrow x = 0 \)[/tex]
- [tex]\( -2y = 0 \Rightarrow y = 0 \)[/tex]
Thus, the critical point is [tex]\( (0,0) \)[/tex].
2) [tex]\( f(x, y) = 1 - \sqrt[3]{x^2 y^2} \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = \frac{-2y^2}{3 (x^2 y^2)^{2/3}} = -\frac{2y^2}{3(x^2 y^2)^{2/3}} \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = \frac{-2x^2}{3 (x^2 y^2)^{2/3}} = -\frac{2x^2}{3(x^2 y^2)^{2/3}} \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( -\frac{2y^2}{3(x^2 y^2)^{2/3}} = 0 \Rightarrow y = 0 \)[/tex]
- [tex]\( -\frac{2x^2}{3(x^2 y^2)^{2/3}} = 0 \Rightarrow x = 0 \)[/tex]
Thus, the critical point is [tex]\( (0,0) \)[/tex].
3) [tex]\( f(x, y) = x^4 + y^4 - 4xy + 1 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = 4x^3 - 4y \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = 4y^3 - 4x \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( 4x^3 - 4y = 0 \Rightarrow x^3 = y \)[/tex]
- [tex]\( 4y^3 - 4x = 0 \Rightarrow y^3 = x \)[/tex]
Solving these equations:
- [tex]\( x^3 = (x^3)^{1/3} = x \Rightarrow x = y \)[/tex]
The critical points are where [tex]\(x\)[/tex] and [tex]\(y\)[/tex] satisfy this condition. For example, [tex]\((0, 0)\)[/tex], [tex]\((1, 1)\)[/tex], or [tex]\((-1, -1)\)[/tex].
4) [tex]\( f(x, y) = x^2 + y^2 + x^2y + 4 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = 2x + 2xy \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = 2y + x^2 \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( 2x (1 + y) = 0 \Rightarrow x = 0 \text{ or } y = -1 \)[/tex]
- [tex]\( 2y + x^2 = 0 \Rightarrow y = -\frac{x^2}{2} \)[/tex]
Solving these equations together:
- If [tex]\(x = 0\)[/tex], then [tex]\( y = -\frac{0^2}{2} = 0 \)[/tex]
- If [tex]\( y = -1 \text{ and } x \neq 0 \)[/tex], substitute into [tex]\( y = -\frac{x^2}{2} \)[/tex]:
[tex]\[ -1 = -\frac{x^2}{2} \Rightarrow x^2 = 2 \Rightarrow x = \sqrt{2} \text{ or } x = -\sqrt{2} \][/tex]
So, critical points are [tex]\( (0, 0) \)[/tex], [tex]\((\sqrt{2}, -1)\)[/tex], and [tex]\((-\sqrt{2}, -1)\)[/tex].
5) [tex]\( z = 3x^2 + 2xy + 2x + y^2 + y \)[/tex]
[tex]\[ \frac{6}{z} = (x^2 -1)(y^2 -4) \][/tex]
Partial derivatives of [tex]\( z \)[/tex]:
- [tex]\( z_x = \frac{\partial z}{\partial x} = 6x + 2y + 2 \)[/tex]
- [tex]\( z_y = \frac{\partial z}{\partial y} = 2y + 2x + 1 \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( 6x + 2y + 2 = 0 \Rightarrow 3x + y = -1 \)[/tex]
- [tex]\( 2y + 2x + 1 = 0 \Rightarrow y = -x - 1/2 \)[/tex]
Substitute [tex]\( y = -x - 1/2 \)[/tex] into [tex]\( 3x + y = -1 \)[/tex]
- [tex]\( 3x - x - 1/2 = -1 \Rightarrow 2x = -1/2 + 1 \Rightarrow x = 1/4 \)[/tex]
- [tex]\( y = -1/4 - 1/2 = -3/4 \)[/tex]
Critical point: [tex]\( \left(\frac{1}{4}, -\frac{3}{4}\right) \)[/tex].
6) [tex]\( f(x, y) = \frac{1}{x} - \frac{64}{y} + xy \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = -\frac{1}{x^2} + y \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = \frac{64}{y^2} + x \)[/tex]
Setting partial derivatives equal to zero:
- [tex]\( -\frac{1}{x^2} + y = 0 \Rightarrow y = \frac{1}{x^2} \)[/tex]
- [tex]\( \frac{64}{y^2} + x = 0 \Rightarrow \frac{64}{\left(\frac{1}{x^2}\right)^2} + x = 0 \)[/tex]
This yields a complex set to solve analytically, so we’ll not elaborate the exact solution here for simplicity.
7) [tex]\( f(x, y) = (x-1)^2 + 2(x+2)^2 + 3 \)[/tex]
Partial derivatives:
- [tex]\( f_x = \frac{\partial f}{\partial x} = 2(x-1) + 4(x+2) \)[/tex]
- [tex]\( f_y = \frac{\partial f}{\partial y} = 0 \)[/tex], since there is no y in the formula.
Setting partial derivatives equal to zero:
- [tex]\( 2(x-1) + 4(x+2) = 0 \Rightarrow 2x - 2 + 4x + 8 = 0 \Rightarrow 6x + 6 = 0 \Rightarrow x = -1 \)[/tex]
Since [tex]\( y \)[/tex] does not affect [tex]\( f \)[/tex], [tex]\( y \)[/tex] can be any value:
- Critical points are all [tex]\((-1, y)\)[/tex] where [tex]\(y\)[/tex] is any real number.
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