Max. and Min. value

[man111]

Find Max. and Min. value of [tex]f(x)=\sqrt{x^3-6x^2+21x+18}[/tex] Where [tex]x,y\in R[/tex] and [tex]-\frac{1}{2}\leq x\leq 1.[/tex]

[ins-]

Read this article:
http://library.thinkquest.org/3616/Calc/S2/FMME.html
You can use it for your problem. Just keep in mind that the expression under the radical sign should be greater than or equal to zero.

[mkmarinov]

Since the square root function is increasing, we shall only analyse the cubic function - the extrema for the sqrt are for the same values of x.
Let [tex]g(x)=x^3-6x^2+21x+18[/tex]
[tex]g'(x)=3x^2-12x+21=3(x^2-4x+7)=3(x^2-4x+4+3)=3((x-2)^2+3)>0[/tex]. Therefore g(x) is increasing and since [tex]f(x)=\sqrt{g(x)}[/tex], f(x) is also increasing. It is easily checked that for all x in the given interval, f(x) is defined.

[man111]

Thanks Mkmarinov.