1. a) For which values of $m$ and $n$ does the expression

$$
\frac{1}{4 m^{2}+12 m n+9 n^{2}+1}
$$

have the greatest value?
b) Sketch the graph of the function

$$
y=(4 m+6 n-2)|x|+2 m+3 n
$$

for all values of $m$ and $n$ for which the expression $\frac{1}{4 m^{2}+12 m n+9 n^{2}+1}$ has the greatest value.
Solution. a) The expression

$$
\frac{1}{4 m^{2}+12 m n+9 n^{2}+1}
$$

has the largest value if

$$
4 m^{2}+12 m n+9 n^{2}+1
$$

has the smallest value.

Since

$$
4 m^{2}+12 m n+9 n^{2}+1=(2 m+3 n)^{2}+1 \geq 1
$$

![](https://cdn.mathpix.com/cropped/2024_06_05_5cbf21d72fe8685988dag-04.jpg?height=353&width=589&top_left_y=436&top_left_x=871)

the smallest value of $4 m^{2}+12 m n+9 n^{2}+1$ is obtained if $2 m+3 n=0$.

b) Since $2 m+3 n=0$, we get that the function is $y=-2|x|$ and its graph is shown in the above diagram.