- #1
squenshl
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- 4
Homework Statement
1. Maximise ##x_1^2+(x_2-5)^2## subject to ##x_1 \geq 0, x_2 \geq 0## and ##2x_1+x_2 \leq 4##.
2. Minimise ##x_1^2+(x_2-5)^2## subject to ##x_1 \geq 0, x_2 \geq 0## and ##2x_1+x_2 \leq 4##.
3. Maximise ##2x_2^2-x_1## subject to ##x_1 \geq 0, x_2 \geq 0## and ##x_1^2+x_2^2 \leq 1##.
Homework Equations
The Attempt at a Solution
1. After doing the process of Lagrange multipliers I get ##x_1 = \lambda## and ##x_2 = \frac{\lambda+10}{2}## which in turn gives ##\lambda = -\frac{2}{5}## meaning ##x_1 = -\frac{2}{5}## and ##x_2 = \frac{24}{5}##. But since ##x_1, \lambda < 0## which don't satisfy ##x_1 \geq 0## and Kuhn-Tucker conditions I'm not sure what to do next.
2. After doing the process of Lagrange multipliers I get ##x_1 = -\lambda## and ##x_2 = \frac{10-\lambda}{2}## which in turn gives ##\lambda = \frac{2}{5}## meaning ##x_1 = -\frac{2}{5}## and ##x_2 = \frac{24}{5}##. But since ##x_1, \lambda < 0## which don't satisfy ##x_1 \geq 0## and Kuhn-Tucker conditions I'm not sure what to do next.
3. After doing the process of Lagrange multipliers I get ##x_1 = -\frac{1}{2\lambda}## and ##x_2 = 0## which in turn gives ##\lambda = 2## meaning ##x_1 = -\frac{1}{4}##. But since ##x_1 < 0## which don't satisfy ##x_1 \geq 0## and Kuhn-Tucker conditions I'm not sure what to do next.