# Thread: maximum value of func.

1. ## maximum value of func.

$\displaystyle f(x)=2(a-x)(\sqrt{x^2+b^2} + x)$

for all real x,
show the maximum value of f(x) is $\displaystyle ( a^2+b^2)$
and $\displaystyle x=\frac{a^2-b^2}{2a}$ when f(x) is maximum

2. Hi stud_02!

Originally Posted by stud_02
$\displaystyle f(x)=2(a-x)(\sqrt{x^2+b^2} + x)$

for all real x,
show the maximum value of f(x) is $\displaystyle ( a^2+b^2)$
and $\displaystyle x=\frac{a^2-b^2}{2a}$ when f(x) is maximum
For all real x? Your Solution seems to be wrong,
if a=0 it does not make any sense

And by the way, if b = 1 and a = 0
$\displaystyle f(x) = -2x(\sqrt{x^2+1}+x)$ has its maximum in $\displaystyle \approx (-0.34 ; -1.66)$

3. Originally Posted by stud_02
$\displaystyle f(x)=2(a-x)(\sqrt{x^2+b^2} + x)$

for all real x,
show the maximum value of f(x) is $\displaystyle ( a^2+b^2)$
and $\displaystyle x=\frac{a^2-b^2}{2a}$ when f(x) is maximum
Substitute
$\displaystyle t = \sqrt{x^2+b^2} + x$
$\displaystyle \implies$
$\displaystyle t^2 - 2tx + x^2 = x^2 + b^2$
$\displaystyle \implies$

$\displaystyle x = \frac{t^2-b^2}{2t}$

Go back to the function ,

we will have
$\displaystyle f(t) = -t^2 + 2at + b^2$

$\displaystyle = -(t-a)^2 + b^2 + a^2$

so it has max. value $\displaystyle a^2 + b^2$ when $\displaystyle t = a$
$\displaystyle x = \frac{a^2-b^2}{2a}$