Find the maximum

Note that $a^2+b^2 = a^2b^2$ $\implies \color{#3D99F6} \dfrac 1{b^2} + \dfrac 1{a^2} = 1$ and $\dfrac {12a+9b+ab}{ab} = {\color{#D61F06}\dfrac {12}b + \dfrac 9a} + 1$ .

Using Cauchy-Schwarz inequality :

$\begin{aligned} \left({\color{#D61F06}\frac {12}b + \frac 9a}\right)^2 & \le \left(12^2 + 9^2 \right) \left(\color{#3D99F6} \dfrac 1{b^2} + \dfrac 1{a^2} \right) = 225({\color{#3D99F6}1}) \\ {\color{#D61F06}\frac {12}b + \frac 9a} & \le 15 \end{aligned}$

Equality occurs when $a = \dfrac 53$ and $b=\dfrac 54$ .

$\begin{aligned} \implies \frac {12a+9b+ab}{ab} & = {\color{#D61F06}\dfrac {12}b + \dfrac 9a} + 1 \le {\color{#D61F06}15}+1 = 16 \end{aligned}$

$\implies n = \boxed{16}$

Note that $a^{2} + b^{2} = a^{2}b^{2}$ $\Rightarrow \frac{1}{a^{2}} + \frac{1}{b^{2}} = 1$ .

Now, put $\frac{1}{a} = sin(A)$ and $\frac{1}{b} = sin(B)$

$\Rightarrow \frac{12a + 9b + ab}{ab} = 12sin(A) + 9sin(B) +1$

Now maximum value of this expression will be $\sqrt{(12)^{2} + (9)^{2}} + 1 = 15 +1 =$ $\boxed{16}$

$a^2+b^2=a^2b^2 \\ \implies \dfrac{a^2+b^2}{a^2b^2}=1 \\ \implies \dfrac{1}{a^2} + \dfrac{1}{b^2}=1$

As $\dfrac{1}{a^2} , \dfrac{1}{b^2} \leq 1$ .So $\dfrac{1}{a^2}=sin^2x, \dfrac{1}{b^2}=cos^2x$

$sinx=\dfrac{1}{a}, cosx=\dfrac{1}{b}$

The above expression becomes $\dfrac{24}{b} + \dfrac{7}{a}+1 \\=7sinx+24cosx+1$

$\sqrt{24^2+7^2}( \dfrac{7}{\sqrt{24^2+7^2}}sinx + \dfrac{24}{\sqrt{24^2+7^2}}cosx)+1$

$=25sin(x+ \tan^{-1}(\dfrac{24}{7})) + 1$

So, the maximum value is $25 \times 1+1=26$ and minimum value is $25(-1)+1=-24$

$\dfrac{24a + 7b + ab}{ab} \\ = \dfrac{24}{b} + \dfrac{7}{a} + 1 \\ \text{Now using Cauchy} {\left( \dfrac{24}{b} + \dfrac{7}{a} \right)}^2 \le {\left(24^2 + 7^2 \right)}{\color{#3D99F6}\left(\frac{a^2+b^2}{a^2 b^2} \right)} \\ \implies {\left(\dfrac{24}{b} + \dfrac{7}{a} \right)} \le 25 \\ \implies \dfrac{24a + 7b + ab}{ab} \le 25 + 1 \\ \quad \quad \le 26$

$\text{since} {\color{#3D99F6}\left(\frac{a^2+b^2}{a^2 b^2} \right)} = \color{#3D99F6}1$

Hence the maximum value is 26

Saya menggunakan a=b dan saya membulatkannya ke bilangan bulat terdekat dan saya mendapatkan hasilnya 16