Maxima and Minima of a function

[n0_3sc]

Are there any analytical techniques to do this besides the Derivative Test?

[HallsofIvy]

There is no general method except by checking where the derivitive is 0 (or does not exist). For some functions, there are other ways. For example we can always find minima and maxima for quadratic functions by completing the square.

[n0_3sc]

I see. Thanks for that.

[lukaszh]

But there is also possibility to estimate. If you solve some elementary function, for example:
f(x)=x^2+3x+2
You can transform it to form:
f(x)+\frac{1}{4}=\left(x+\frac{3}{2}\right)^2
So now you are able to find a minimum:
\min_{x\in\mathbb{R}}f(x)=-\frac{1}{4}

[n0_3sc]

Yes, but my function is far too complex/tedious to do either way. An expression for the min and max has been found though proving it is too difficult for me.

[NoMoreExams]

But there is also possibility to estimate. If you solve some elementary function, for example:
f(x)=x^2+3x+2
You can transform it to form:
f(x)+\frac{1}{4}=\left(x+\frac{3}{2}\right)^2
So now you are able to find a minimum:
\min_{x\in\mathbb{R}}f(x)=-\frac{1}{4}

Well if you are working with quadratics a lot you should know that the min/max is going to be at \left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)