> 
> Find the maximum value of $\sqrt{x - 144} + \sqrt{722 - x} .$
> 
> 
> 

**What I Tried** :- Let me tell you first . What I Tried is absolutely silly , but you may check for it .

I thought AM-GM would do the trick and got :-

$\sqrt{x - 144} + \sqrt{722 - x} \geq 2 \sqrt{\sqrt{(x - 144)(722 - x)}}$

$\rightarrow \sqrt{x - 144} + \sqrt{722 - x} \geq 2 \sqrt{\sqrt{(-x^2 + 866x + 103968)}}$

From here I realised that there's no useful information I can find for $x$ .

**Back to Another Attempt** :- let $P = \sqrt{x - 144} + \sqrt{722 - x}$ . Then :-

$P^2 = 578 + 2\sqrt{(x - 144)(722 - x)}$

As $P \geq 0$ , we have that $P^2 \geq 578 \rightarrow P \geq 17\sqrt2 .$

This Attempt seemed reasonable and I thought I already found the solution, but then I realised that this is the minimum value of $P$ , and now I am hopeless .

Wolfram Alpha gives the answer to be $34$ , which can also be checked by Trial and Error of integer values , but that does not seem to be a proof of this problem .

Can anyone help me with this?
['https://math.stackexchange.com/questions/3814894', 'https://math.stackexchange.com', 'https://math.stackexchange.com/users/772237/']
I agree with Evariste's answer, but would like to offer an alternative approach. My approach **begins** with Evariste's conclusion that it is desired to maximize

$(x - 144)(722 - x) = -x^2 + x(866) - (144 \times 722).$

This is equivalent to trying to maximize

$-x^2 + x(866)$, where (presumably) $x$ is required to be in the interval [144, 722].

Suppose you pose the equation $-x^2 + x(866) = k.$   

Then the question is what is the largest possible (real) value for $k$   

that will generate at least one **real** root for $x.$

The above equation is equivalent to the equation $x^2 - x(866) + k = 0.$

This equation will have at least one real root if and only if   

$[(866)^2 - 4k] \;\geq\; 0.$

This means that the largest permissible value of $k$ is   

$\frac{1}{4} \times (866)^2.$

It is immediate that choosing this value for $k$ will cause   

$[(866)^2 - 4k]$ to equal 0.

This means that with this choice of $k$, the root(s) of   

$x^2 - x(866) + k = 0$ will be   

$x = (866/2)$.

By the above analysis, the OP's original expression should therefore be maximized by   

$x = 433.$
By C-S $$\sqrt{x-144}+\sqrt{722-x}\leq\sqrt{(1+1)(x-144+722-x)}=34.$$
The equality occurs for $(1,1)||(x-144,722-x),$ which says that we got a maximal value.