Maximum value of $8v_1 - 6v_2 - v_1^2 - v_2^2$ subject to $v_1^2+v_2^2\leq 1$ 
Given that $g:\mathbb{R}^2 \to \mathbb{R}$ defined by 
  $$g(v_1,v_2) = 8v_1 - 6v_2 - v_1^2 - v_2^2$$
  find the maximum value of $g$ subject to the constraint $v_1^2+v_2^2\leq 1.$

My attempt: 
Note that 
$$g(v_1,v_2) = 8v_1 - 6v_2 - v_1^2 - v_2^2 = -(v_1-4)^2 - (v_2+3)^2 + 25.$$
So $g$ is a decreasing function.
So, the maximum value of $g$ lies on the circumference of $v_1^2+v_2^2 = 1.$
It suffices to find the intersection between $(0,0)$ and $(4,-3)$ as $(4,-3)$ is the peak point of $g.$
The intersection point lies on both $v_1^2+v_2^2 = 1$ and $v_2 = -\frac{3}{4}v_1.$
Solving the simultaneous equation gives that 
$$v_1 = \frac{4}{5}, \quad v_2 = -\frac{3}{5}.$$
So, maximum value of $g$ is 
$$g\left(\frac{4}{5}, -\frac{3}{5}\right) = 9.$$
Is my attempt correct?

Your result is correct but it is not clear what you mean by $g$ is a decreasing function and "find the intersection between $(0,0)$ etc."
So, here is another way using Cauchy-Schwarz:
$$8v_1 - 6v_2 \leq \sqrt{8^2+6^2}\sqrt{v_1^2+v_2^2} = 10 \sqrt{v_1^2+v_2^2}$$
$$g(v_1,v_2)\leq 10\sqrt{v_1^2+v_2^2}- (v_1^2 + v_2^2)=t(10-t) \mbox{ with } t = \sqrt{v_1^2+v_2^2}$$
$t(10-t)$ is strictly increasing for $0\leq t \leq 5 \Rightarrow$ maximum of $g$ is reached for $t=\sqrt{v_1^2+v_2^2} =1$: $1(10-1)=\boxed{9}$
