Equality Constraint Optimizations

Question

I have an optimization problem of this sort

$min_{(x_1, x_2) } 4x^2_1 + x^2_2$

subject to $x_1 -2x_2 + 5 = 0$

I am trying to solve for the optimal points using the first and second derivative tests by making $x_1$ function of $x_2$.

I have that $x_1 = 2x_2 +5$ and the minimization problem is of the form $17x_2^2 - 80x_2 +100$ From this point I'm guessing that first derivative of $17x_2^2 - 80x_2 +100$ =0. Hence $x_2 = 80/34$ I'm not sure if this is correct or how to do the second derivative test.

Answer 1

We have $x_1 = 2x_2\color{red}-5$

$$4(2x_2-5)^2+x_2^2=17x_2^2-80x_2+100$$

Setting the first derivative $34x_2-80=0$.

The second derivative is $17 > 0$, hence it is the minimum.

Answer 2

Another way, without derivative -

$x_1 = 2x_2 - 5$

So, $4x_1^2+x_2^2 = 4(4x_2^2-20x_2+25)+x_2^2 = 17x_2^2-80x_2+100$

$ = 17(x_2^2 - \frac{80}{17}x_2+\frac{100}{17}) = 17[(x_2-\frac{40}{17})^2 + \frac{100}{17^2}$.

Min of square will be zero so min value of $4x_1^2+x_2^2$ is $\frac{100}{17}$ when $x_2 = \frac{40}{17}$. Can also get the value of $x_1$.