Maxima and Minima of a function

by n0_3sc
Tags: function, maxima, minima
 P: 265 Are there any analytical techniques to do this besides the Derivative Test?
 PF Patron Sci Advisor Thanks Emeritus P: 38,429 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.
 P: 265 I see. Thanks for that.
P: 32

Maxima and Minima of a function

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:
$$\min_{x\in\mathbb{R}}f(x)=-\frac{1}{4}$$
 P: 265 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.
P: 626
 Quote by 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: $$\min_{x\in\mathbb{R}}f(x)=-\frac{1}{4}$$
Well if you are working with quadratics a lot you should know that the min/max is going to be at $$\left(\frac{-b}{2a}, f\left(\frac{-b}{2a}\right)\right)$$

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