Given that $|3x|+|2y| \leq 1$ then the maximum value of $9x+4y$ is ?.

Question

Suppose that $|3x|+|2y| \leq 1$ then the maximum value of $9x+4y$ is?.

I thought of drawing the region satisfied the constraint given on the $xy$ plane. here is the region enclosed the lines -

I recall a theorem stating that maximum value of the function under the given constraints occurs at the boundaries. Is this correct?. I tried to substitute the following four points into the equation $9x+4y$ and got the maximum value as $3$ and I think the minimum value is $-3$ similarly. Is this correct?

Any other way or approach of solving this?

Answer 1

$9x+4y \le |9x+4y| \le 2|3x|+2|2y|+|3x| \le 2+|3x| \le 2+1=3$ and for $x=1/3$ and $y=0$ we have $9x+4y=3$.

Answer 2

The gradient of $9x+4y$ is $(9,4)$. Move along that from $-\infty$ to reach the constraint, pass through it and exit from the other side. The last point visited by this process is the maximal point which is $(\dfrac{1}{3},0)$ and the maximum is $3$

Here is a sketch