Math / Calculus

Question2. Исследовать функции на условный экстремум. (по 1 баллу) $2.1 z=\sqrt[4]{x} \sqrt[3]{y} \quad$ при условии $2 x+5 y=100$

Studdy Solution

STEP 1

1. We are given a function $z = \sqrt[4]{x} \sqrt[3]{y}$ and a constraint $2x + 5y = 100$ .
2. The problem requires finding conditional extrema, which suggests using methods such as Lagrange multipliers.
3. The variables $x$ and $y$ must be positive for the roots to be real.

STEP 2

1. Set up the Lagrange function.
2. Find the partial derivatives of the Lagrange function.
3. Solve the system of equations derived from the partial derivatives.
4. Verify the solutions satisfy the constraint.
5. Determine the nature of the extrema.

STEP 3

Set up the Lagrange function $\mathcal{L}(x, y, \lambda)$ by incorporating the constraint into the objective function using a Lagrange multiplier $\lambda$ :
$\mathcal{L}(x, y, \lambda) = \sqrt[4]{x} \sqrt[3]{y} + \lambda (2x + 5y - 100)$

STEP 4

Find the partial derivatives of the Lagrange function with respect to $x$ , $y$ , and $\lambda$ :
$\frac{\partial \mathcal{L}}{\partial x} = \frac{1}{4}x^{-3/4}y^{1/3} + 2\lambda$ $\frac{\partial \mathcal{L}}{\partial y} = \frac{1}{3}x^{1/4}y^{-2/3} + 5\lambda$ $\frac{\partial \mathcal{L}}{\partial \lambda} = 2x + 5y - 100$

STEP 5

Set each of the partial derivatives to zero to form a system of equations:
$\frac{1}{4}x^{-3/4}y^{1/3} + 2\lambda = 0$ $\frac{1}{3}x^{1/4}y^{-2/3} + 5\lambda = 0$ $2x + 5y - 100 = 0$
Solve this system to find values of $x$ , $y$ , and $\lambda$ .

STEP 6

From the first two equations, express $\lambda$ in terms of $x$ and $y$ :
$\lambda = -\frac{1}{8}x^{-3/4}y^{1/3}$ $\lambda = -\frac{1}{15}x^{1/4}y^{-2/3}$
Equate the two expressions for $\lambda$ and solve for the relationship between $x$ and $y$ .

STEP 7

Equating the expressions for $\lambda$ :
$-\frac{1}{8}x^{-3/4}y^{1/3} = -\frac{1}{15}x^{1/4}y^{-2/3}$
Simplify and solve for $x$ in terms of $y$ or vice versa.

STEP 8

Solve the simplified equation from Step 5 and use it with the constraint $2x + 5y = 100$ to find specific values for $x$ and $y$ .

STEP 9

Verify that the solutions for $x$ and $y$ satisfy the original constraint $2x + 5y = 100$ .

STEP 10

Determine the nature of the extrema (whether they are maxima, minima, or saddle points) by evaluating the second derivatives or using other criteria for conditional extrema.
The solution involves solving the system of equations and verifying the constraint, leading to specific values for $x$ and $y$ that optimize the function $z$ under the given constraint.

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