Question

Найти наименьшее значение выражения 3*a^2 + 5ab - 2*b^2, если b - 2a = 2.

Answer 1

Из уравнения b - 2a = 2 выражаем b:
b = 2 + 2a, подставим это в выражение:

3a² + 5a*(2+2a) - 2*(2+2a)² = 3a² + 10a + 10a² - 2*(4+8a+4a²) = 3a² + 10a + 10a² - 8 - 16a - 8a² = 5a² - 6a - 8 = 5*(a² - 6/5a) - 8 = 5*(a² - 2*3/5) + 9/5 - 49/5 =5*(a² - 2*3/5 + 9/25) - 49/5 = - 5*(a-3/5)² - 49/5

Наименьшего значения будет в точке а=3/5, равное -49/5

Answer 2

To find the minimum value of the expression $3a^2 + 5ab - 2b^2$ under the given constraint $b - 2a = 2$, we can use the method of Lagrange multipliers. First, we define the function $f(a, b) = 3a^2 + 5ab - 2b^2$ and the constraint function $g(a, b) = b - 2a - 2$.

Now, we form the Lagrangian function $L(a, b, \lambda) = f(a, b) + \lambda g(a, b)$, where $\lambda$ is the Lagrange multiplier.

The partial derivatives are: $\frac{\partial L}{\partial a} = 6a + 5b - 4\lambda$, $\frac{\partial L}{\partial b} = 5a - 4b + \lambda$, $\frac{\partial L}{\partial \lambda} = b - 2a - 2$.

To find the critical points, we set the partial derivatives equal to zero:

$6a + 5b - 4\lambda = 0$ ...(i) $5a - 4b + \lambda = 0$ ...(ii) $b - 2a - 2 = 0$ ...(iii)

Now we can solve these three equations simultaneously to find the values of $a$, $b$, and $\lambda$.

From equation (iii), we have $b = 2a + 2$. Now, we can substitute this value of $b$ into equation (ii):

$5a - 4(2a + 2) + \lambda = 0$

$5a - 8a - 8 + \lambda = 0$

$-3a - 8 + \lambda = 0$

$\lambda = 3a + 8$ ...(iv)

Next, we substitute the value of $\lambda$ from equation (iv) into equation (i):

$6a + 5(2a + 2) - 4(3a + 8) = 0$

$6a + 10a + 10 - 12a - 32 = 0$

$4a - 22 = 0$

$4a = 22$

$a = \frac{22}{4} = \frac{11}{2} = 5.5$

Now, we can find the value of $b$ using the value of $a$ from above:

$b = 2a + 2 = 2 \times 5.5 + 2 = 11 + 2 = 13$

Finally, we can find the minimum value of the expression by substituting the values of $a$ and $b$ into the function $f(a, b) = 3a^2 + 5ab - 2b^2$:

$f\left(\frac{11}{2}, 13\right) = 3\left(\frac{11}{2}\right)^2 + 5\left(\frac{11}{2}\right) \times 13 - 2 \times 13^2$

$f\left(\frac{11}{2}, 13\right) = \frac{363}{2} + \frac{715}{2} - 338$

$f\left(\frac{11}{2}, 13\right) = \frac{1072}{2} - 338$

$f\left(\frac{11}{2}, 13\right) = 536 - 338$