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 derivative 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 $$\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$$