How to Find the Maximum of the Function f(x)=12x-32+24√(9-3x), for x≤3?

[Albert1]

$f(x)=12x-32+24\sqrt {9-3x}, x\leq 3$
find $max f(x)$

 

[kaliprasad]

$f(x)=12x-32+24\sqrt {9-3x}, x\leq 3$
find $max f(x)$

Spoiler

let 9-3x = y^2 so 3x = 9-y^2 so $f(x) = 36-4y^2 - 32 + 24y = 4 - (4y^2 - 24 y) = 4 - 4(y- 3)^2 + 36 = 40 - 4(y-3)^2$
lowest when y = 3 ( range of y is >0 feasible and x = 0 so i we get 40 as maximum $f(x)$